Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

12-1. Finding Limits: Numerical and Graphical Approaches

Finding Limits: Numerical and Graphical Approaches
In this section, you will:
  • Understand limit notation.
  • Find a limit using a graph.
  • Find a limit using a table.

Intuitively, we know what a limit is. A car can go only so fast and no faster. A trash can might hold 33 gallons and no more. It is natural for measured amounts to have limits. What, for instance, is the limit to the height of a woman? The tallest woman on record was Jinlian Zeng from China, who was 8 ft 1 in.1 Is this the limit of the height to which women can grow? Perhaps not, but there is likely a limit that we might describe in inches if we were able to determine what it was.

To put it mathematically, the function whose input is a woman and whose output is a measured height in inches has a limit. In this section, we will examine numerical and graphical approaches to identifying limits.

Understanding Limit Notation

We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>,</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 , 1 4 , 1 8 ...

gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=L,  then as the input<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>gets closer and closer to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, the output y-coordinate gets closer and closer to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.  We say that the output “approaches”<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.

[link] provides a visual representation of the mathematical concept of limit. As the input value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mi>x</mi><mtext>  </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, the output value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.

<figure class="small" id="CNX_Precalc_Figure_12_01_001"> <figcaption>The output (y--coordinate) approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as the input (x-coordinate) approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.</figcaption> Graph representing how a function with a hole at (a, L) approaches a limit.</figure>

We write the equation of a limit as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→a f(x)=L.

This notation indicates that as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mi>a</mi><mtext>  </mtext></annotation-xml></semantics></math>both from the left of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow></annotation-xml></semantics></math>  and the right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mo>,</mo></mrow></annotation-xml></semantics></math> the output value approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.

Consider the function

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 −6x−7 x−7 .

We can factor the function as shown.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> (x−7) (x+1) x−7   Cancel like factors in numerator and denominator. f(x)=x+1,x≠7 Simplify.

Notice that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mrow><mi>x</mi><mo>≠</mo><mn>7</mn><mo>,</mo></mrow></annotation-xml></semantics></math> for the simplified function. We can represent the function graphically as shown in [link].

<figure class="small" id="CNX_Precalc_Figure_12_01_002"> <figcaption>Because 7 is not allowed as an input, there is no point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=7.</figcaption> Graph of an increasing function, f(x) = (x^2-6x-7)/(x-7), with a hole at (7, 8).</figure>

What happens at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=7  is completely different from what happens at points close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=7  on either side. The notation

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 7 f(x)=8

indicates that as the input<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7.

What happens at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=7?  When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=7, there is no corresponding output. We write this as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mtext> does not exist</mtext><mtext>.</mtext></mrow></annotation-xml></semantics></math>

This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>,</mo><mtext>  </mtext><mi>x</mi><mo>≠</mo><mn>7.</mn></mrow></annotation-xml></semantics></math>

Notice that the limit of a function can exist even when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  is not defined at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  Much of our subsequent work will be determining limits of functions as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>nears<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, even though the output at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  does not exist.

The Limit of a Function

A quantity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>is the limit of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>if, as the input values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>(but do not equal<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a),the corresponding output values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  get closer to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.  Note that the value of the limit is not affected by the output value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  Both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>must be real numbers. We write it as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→a f(x)=L
Understanding the Limit of a Function

For the following limit, define<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a,f(x),  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→2  ( 3x+5 )=11

First, we recognize the notation of a limit. If the limit exists, as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, we write

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→a  f(x)=L.

We are given

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→2 ( 3x+5 )=11.

This means that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a=2,f(x)=3x+5, and L=11.

Analysis

Recall that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=3x+5  is a line with no breaks. As the input values approach 2, the output values will get close to 11. This may be phrased with the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→2 (3x+5)=11 , which means that as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>nears 2 (but is not exactly 2), the output of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=3x+5  gets as close as we want to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 3(2)+5, or 11, which is the limit<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L, as we take values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>sufficiently near 2 but not at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.

For the following limit, define<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a,f(x),and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→5 ( 2 x 2 −4 )=46

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>a</mi><mo>=</mo><mn>5</mn><mo>,</mo></mrow></annotation-xml></semantics></math><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )=2 x 2 −4,and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L=46.

Understanding Left-Hand Limits and Right-Hand Limits

We can approach the input of a function from either side of a value—from the left or the right. [link] shows the values of

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>x</mi><mo>≠</mo><mn>7</mn></mrow></annotation-xml></semantics></math>

as described earlier and depicted in [link].

<figure id="CNX_Precalc_Figure_12_01_013">Table showing that f(x) approaches 8 from either side as x approaches 7 from either side.</figure>

Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in [link] are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 6.9,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 6.99, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 6.999.  The corresponding outputs are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 7.9,7.99, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 7.999.  These values are getting closer to 8. The limit of values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=x+1,x≠7  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 7.

Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in [link] are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 7.1,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 7.01,and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 7.001.  The corresponding outputs are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 8.1,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 8.01, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext>  </mtext><mrow><mn>8.001.</mn></mrow></annotation-xml></semantics></math>  These values are getting closer to 8. The limit of values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=x+1,x≠7  as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 7.

[link] shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>within the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 6.9<x<7.1  to produce an output value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  within the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 7.9<f(x)<8.1.

We also see that we can get output values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.

[link] provides a visual representation of the left- and right-hand limits of the function. From the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x), we observe the output can get infinitesimally close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L=8  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 7 from the left and as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 7 from the right.

To indicate the left-hand limit, we write

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 7 − f(x)=8.

To indicate the right-hand limit, we write

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 7 + f(x)=8.
<figure class="small" id="CNX_Precalc_Figure_12_01_003"> <figcaption>The left- and right-hand limits are the same for this function.</figcaption> Graph of the previous function explaining the function's limit at (7, 8)</figure>
Left- and Right-Hand Limits

The left-hand limit of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>from the left is equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L, denoted by

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ a − f(x)=L.

The values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  can get as close to the limit<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as we like by taking values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>sufficiently close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x<a  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x≠a.

The right-hand limit of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x), as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>from the right, is equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L,denoted by

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ a + f(x)=L.

The values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  can get as close to the limit<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as we like by taking values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>sufficiently close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>but greater than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  Both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>are real numbers.

Understanding Two-Sided Limits

In the previous example, the left-hand limit and right-hand limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>are equal. If the left- and right-hand limits are equal, we say that the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  has a two-sided limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

The Two-Sided Limit of Function as x Approaches a

The limit of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x), as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, is equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L, that is,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→a  f(x)=L

if and only if

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ a − f(x)= lim x→ a + f(x).

In other words, the left-hand limit of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>is equal to the right-hand limit of the same function as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.

Finding a Limit Using a Graph

To visually determine if a limit exists as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, we observe the graph of the function when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>is very near to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  In[link] we observe the behavior of the graph on both sides of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.

<figure class="small" id="CNX_Precalc_Figure_12_01_004">Graph of a function that explains the behavior of a limit at (a, L) where the function is increasing when x is less than a and decreasing when x is greater than a.</figure>

To determine if a left-hand limit exists, we observe the branch of the graph to the left of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a, but near<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  This is where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x<a.  We see that the outputs are getting close to some real number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>so there is a left-hand limit.

To determine if a right-hand limit exists, observe the branch of the graph to the right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a,but near<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  This is where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x>a.  We see that the outputs are getting close to some real number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L, so there is a right-hand limit.

If the left-hand limit and the right-hand limit are the same, as they are in [link], then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.

Finally, we can look for an output value for the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  when the input value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>is equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  The coordinate pair of the point would be<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( a,f( a ) ).  If such a point exists, then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a )  has a value. If the point does not exist, as in [link], then we say that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a )  does not exist.

Given a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x ),use a graph to find the limits and a function value as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.

  1. Examine the graph to determine whether a left-hand limit exists.
  2. Examine the graph to determine whether a right-hand limit exists.
  3. If the two one-sided limits exist and are equal, then there is a two-sided limit—what we normally call a “limit.”
  4. If there is a point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a, then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a )  is the corresponding function value.
Finding a Limit Using a Graph
  1. Determine the following limits and function value for the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>shown in [link].
    1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 − f(x)
    2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 + f(x)
    3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→2 f(x)
    4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>
    <figure class="small" id="CNX_Precalc_Figure_12_01_005">Graph of a piecewise function that has a positive parabola centered at the origin and goes from negative infinity to (2, 8), an open point, and a decreasing line from (2, 3), a closed point, to positive infinity on the x-axis.</figure>
  2. Determine the following limits and function value for the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>shown in [link].
    1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 − f(x)
    2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 + f(x)
    3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→2 f(x)
    4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>
    <figure class="small" id="CNX_Precalc_Figure_12_01_006">Graph of a piecewise function that has a positive parabola from negative infinity to 2 on the x-axis, a decreasing line from 2 to positive infinity on the x-axis, and a point at (2, 4).</figure>
  1. Looking at [link]:
    1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 − f(x)=8;  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x<2,but infinitesimally close to 2, the output values get close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=8.
    2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2   + f(x)=3;  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x>2,but infinitesimally close to 2, the output values approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=3.
    3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 f(x)  does not exist because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→ 2   − f(x)≠ lim x→ 2   + f(x);  the left and right-hand limits are not equal.
    4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2 )=3  because the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>passes through the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,f( 2 ) )  or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,3 ).
  2. Looking at [link]:
    1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2   − f(x)=8;  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x<2  but infinitesimally close to 2, the output values approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=8.
    2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2   + f(x)=8;  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x>2  but infinitesimally close to 2, the output values approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=8.
    3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 f(x)=8  because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→ 2   − f(x)= lim x→ 2   + f(x)=8;  the left and right-hand limits are equal.
    4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2 )=4  because the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>passes through the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,f( 2 ) )  or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,4 ).

Using the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=f( x )  shown in [link], estimate the following limits.

<figure class="small" id="CNX_Precalc_Figure_12_01_007">Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)</figure>

a. 0; b. 2; c. does not exist; d.<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><mo>;</mo></mrow></annotation-xml></semantics></math>  e. 0; f. does not exist; g. 4; h. 4; i. 4

Finding a Limit Using a Table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>from both sides. Then we determine if the output values get closer and closer to some real value, the limit<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.

Let’s consider an example using the following function:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 5 ( x 3 −125 x−5 )

To create the table, we evaluate the function at values close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=5.  We use some input values less than 5 and some values greater than 5 as in [link]. The table values show that when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x>5  but nearing 5, the corresponding output gets close to 75. When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x>5  but nearing 5, the corresponding output also gets close to 75.

<figure id="CNX_Precalc_Figure_12_01_008">Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.</figure>

Because

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 5 − f(x)=75= lim x→ 5 + f(x),

then

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→5 f(x)=75.

Remember that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( 5 )  does not exist.

Given a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f,use a table to find the limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(a),if it exists.

  1. Choose several input values that approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>from both the left and right. Record them in a table.
  2. Evaluate the function at each input value. Record them in the table.
  3. Determine if the table values indicate a left-hand limit and a right-hand limit.
  4. If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.
  5. Replace<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>to find the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a ).
Finding a Limit Using a Table

Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 ( 5sin(x) 3x )

We can estimate the value of a limit, if it exists, by evaluating the function at values near<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  We cannot find a function value for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0  directly because the result would have a denominator equal to 0, and thus would be undefined.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5sin(x) 3x

We create [link] by choosing several input values close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0, with half of them less than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0  and half of them greater than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.

The table values indicate that when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x<0  but approaching 0, the corresponding output nears<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 5 3 .

When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x>0  but approaching 0, the corresponding output also nears<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 5 3 .

<figure id="CNX_Precalc_Figure_12_01_009">Table shows that as x values approach 0 from the positive or negative direction, f(x) gets very close to 5 over 3. But when x is equal to 0, y is undefined.</figure>

Because

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 0 − f(x)= 5 3 = lim x→ 0 + f(x),

then

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 f(x)= 5 3 .

Is it possible to check our answer using a graphing utility?

Yes. We previously used a table to find a limit of 75 for the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 3 −125 x−5  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 5. To check, we graph the function on a viewing window as shown in [link]. A graphical check shows both branches of the graph of the function get close to the output 75 as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By appraoching <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=5  we may numerically observe the corresponding outputs getting close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 75.

<figure class="small" id="CNX_Precalc_Figure_12_01_010">Graph of an increasing function with a discontinuity at (5, 75)</figure>

Numerically estimate the limit of the following function by making a table:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 ( 20sin(x) 4x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 ( 20sin(x) 4x )=5

Table showing that f(x) approaches 5 from either side as x approaches 0 from either side.

Is one method for determining a limit better than the other?

No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.

Using a Graphing Utility to Determine a Limit

With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0. If the function has a limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0, state it. If not, discuss why there is no limit.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π x )

We can use a graphing utility to investigate the behavior of the graph close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  Centering around<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0, we choose two viewing windows such that the second one is zoomed in closer to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0  than the first one. The result would resemble [link]for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [−2,2]  by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [−3,3].

<figure class="small" id="CNX_Precalc_Figure_12_01_011">Graph of a sinusodial function zoomed in at [-2, 2] by [-3, 3].</figure>

The result would resemble [link] for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [−0.1,0.1]  by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [−3,3].

<figure class="small" id="CNX_Precalc_Figure_12_01_012"> <figcaption>Even closer to zero, we are even less able to distinguish any limits.</figcaption> Graph of the same sinusodial function as in the previous image zoomed in at [-0.1, 0.1] by [-3. 3].</figure>

The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0.

We write

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 0 − ( 3sin( π x ) ) does not exist.
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 0 + ( 3sin( π x ) ) does not exist.
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 ( 3sin( π x ) ) does not exist.

Numerically estimate the following limit:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→0 ( sin( 2 x ) ).

does not exist

Access these online resources for additional instruction and practice with finding limits.

 

Key Concepts

  • A function has a limit if the output values approach some value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as the input values approach some quantity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  See[link].
  • A shorthand notation is used to describe the limit of a function according to the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→ a f(x)=L, which indicates that as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, both from the left of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  and the right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a, the output value gets close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L.
  • A function has a left-hand limit if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x<a.  A function has a right-hand limit if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x>a.
  • A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
  • A graph provides a visual method of determining the limit of a function.
  • If the function has a limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, the branches of the graph will approach the same<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y-coordinate near<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  from the left and the right. See [link].
  • A table can be used to determine if a function has a limit. The table should show input values that approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See [link].
  • A graphing utility can also be used to find a limit. See [link].

Section Exercises

Verbal

Explain the difference between a value at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  and the limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.

The value of the function, the output, at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a ).  When the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→a f( x )  is taken, the values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>get infinitely close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>but never equal<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  As the values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>from the left and right, the limit is the value that the function is approaching.

Explain why we say a function does not have a limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>if, as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a, the left-hand limit is not equal to the right-hand limit.

Graphical

For the following exercises, estimate the functional values and the limits from the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>provided in [link].

<figure class="small" id="CNX_Precalc_Figure_12_01_201">A piecewise function with discontinuities at x = -2, x = 1, and x = 4.</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→− 2 −  f(x)

–4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→− 2 +  f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−2  f(x)

–4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>−2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→− 1 −  f(x)

2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 1 +  f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→1  f(x)

does not exist

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 4 −  f(x)

4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 4 +  f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→4  f(x)

does not exist

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

For the following exercises, draw the graph of a function from the functional values and limits provided.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 0 −  f(x)=2, lim x→ 0 +  f(x)=–3, lim x→2  f(x)=2, f(0)=4, f(2)=–1, f(–3) does not exist.

Answers will vary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 −  f(x)=0,  lim x→ 2 + =–2, lim x→0  f(x)=3, f(2)=5, f(0)

Answers will vary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 −  f(x)=2,  lim x→ 2 +  f(x)=−3,  lim x→0  f(x)=5, f(0)=1, f(1)=0

Answers will vary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 3 −  f(x)=0,  lim x→ 3 +  f(x)=5,  lim x→5  f(x)=0, f(5)=4, f(3) does not exist.

Answers will vary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→4  f(x)=6,  lim x→ 6 +  f(x)=−1,  lim x→0  f(x)=5, f(4)=6, f(2)=6

Answers will vary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−3  f(x)=2,  lim x→ 1 +  f(x)=−2,  lim x→3  f(x)=–4, f(–3)=0, f(0)=0

Answers will vary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→π  f(x)= π 2 ,  lim x→–π  f(x)= π 2 ,  lim x→ 1 –  f(x)=0, f(π)= 2 , f(0) does not exist.

Answers will vary.

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> ( 1+x ) 1 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> ( 1+x ) 2 x

7.38906

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> ( 1+x ) 3 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> ( 1+x ) 4 x

54.59815

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>j</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> ( 1+x ) 5 x

Based on the pattern you observed in the exercises above, make a conjecture as to the limit of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= ( 1+x ) 6 x ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> g(x)= ( 1+x ) 7 x ,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> and h(x)= ( 1+x ) n x .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>e</mi></msup></mrow></annotation-xml></semantics></math> 6 ≈403.428794,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> e 7 ≈1096.633158,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> e n

For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  If the function has a limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a,state it. If not, discuss why there is no limit.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> | x |−1, if x≠1 x 3 , if x=1  a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 1 x+1 , if x=−2 (x+1) 2 , if x≠−2  a=−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−2 f(x)=1

Numeric

For the following exercises, use numerical evidence to determine whether the limit exists at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  If not, describe the behavior of the graph of the function near<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  Round answers to two decimal places.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 −4x 16− x 2 ;a=4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 −x−6 x 2 −9 ;a=3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→3 ( x 2 −x−6 x 2 −9 )= 5 6 ≈0.83

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 −6x−7 x 2 – 7x ;a=7

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 –1 x 2 –3x+2 ;a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→1 ( x 2 −1 x 2 −3x+2 )=−2.00

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1− x 2 x 2 −3x+2 ;a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 10−10 x 2 x 2 −3x+2 ;a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→1 ( 10−10 x 2 x 2 −3x+2 )=20.00

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 6 x 2 −5x−6 ;a= 3 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 4 x 2 +4x+1 ;a=− 1 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ −1 2 ( x 4 x 2 +4x+1 )  does not exist. Function values decrease without bound as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches –0.5 from either left or right.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 x−4 ; a=4

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches the given value.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 7tanx 3x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 7tanx 3x = 7 3

Table shows as the function approaches 0, the value is 7 over 3 but the function is undefined at 0.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→4 x 2 x−4

Table shows as the function approaches 4, the value does not exist since approaching the limit value from the left is negative infinity and approaching the limit value from the right is positive infinity.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 2sinx 4tanx

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 2sinx 4tanx = 1 2

Table shows as the function approaches 0, the value is 1 over 2, but the function is undefined at 0.

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  If the function has a limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a,state it. If not, discuss why there is no limit.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 e e 1 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 e e −  1 x 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 e e −  1 x 2 =1.0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 | x | x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−1 | x+1 | x+1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→− 1 − | x+1 | x+1 = −(x+1) (x+1) =−1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→− 1 + | x+1 | x+1 = (x+1) (x+1) =1;  since the right-hand limit does not equal the left-hand limit,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→−1 | x+1 | x+1  does not exist.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→5 | x−5 | 5−x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−1 1 ( x+1 ) 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−1 1 ( x+1 ) 2  does not exist. The function increases without bound as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> −1  from either side.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→1 1 ( x−1 ) 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 5 1− e 2 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 5 1− e 2 x  does not exist. Function values approach 5 from the left and approach 0 from the right.

Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=| 1−x x |  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> g(x)=| 1+x x |  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> g( x )  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0. If the functions have a limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0, state it. If not, discuss why there is no limit.

Extensions

According to the Theory of Relativity, the mass<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>m</mi><mtext> </mtext></annotation-xml></semantics></math>of a particle depends on its velocity<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>v</mi></annotation-xml></semantics></math>. That is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>m</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> m o 1−( v 2 / c 2 )

where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> m o  is the mass when the particle is at rest and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>c</mi><mtext> </mtext></annotation-xml></semantics></math>is the speed of light. Find the limit of the mass,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> m, as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>v</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> c − .

Through examination of the postulates and an understanding of relativistic physics, as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> v→c,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> m→∞.  Take this one step further to the solution,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> v→ c − m= lim v→ c − m o 1−( v 2 / c 2 ) =∞

Allow the speed of light,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> c, to be equal to 1.0. If the mass,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> m, is 1, what occurs to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>m</mi><mtext> </mtext></annotation-xml></semantics></math>as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> v→c?  Using the values listed in[link], make a conjecture as to what the mass is as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>v</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 1.00.

 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>v</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>m</mi></annotation-xml></semantics></math>
0.5 1.15
0.9 2.29
0.95 3.20
0.99 7.09
0.999 22.36
0.99999 223.61

Glossary

left-hand limit
the limit of values of<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )  as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>the left, denoted<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ a − f(x)=L.  The values of<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>  can get as close to the limit<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as we like by taking values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>sufficiently close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>such that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo><</mo><mi>a</mi></mrow></annotation-xml></semantics></math>  and<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>≠</mo><mi>a</mi><mo>.</mo></mrow></annotation-xml></semantics></math>  Both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>are real numbers.
limit
when it exists, the value,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L,that the output of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  approaches as the input<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>gets closer and closer to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>but does not equal<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  The value of the output,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math>can get as close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as we choose to make it by using input values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>sufficiently near to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a,but not necessarily at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  Both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>are real numbers, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>is denoted<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→a f(x)=L.
right-hand limit
the limit of values of<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )  as<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>from the right, denoted<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ a + f(x)=L.  The values of<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>  can get as close to the limit<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>as we like by taking values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>sufficiently close to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>where<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>></mo><mi>a</mi><mo>,</mo></mrow></annotation-xml></semantics></math>and<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>≠</mo><mi>a</mi><mo>.</mo></mrow></annotation-xml></semantics></math>  Both<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>L</mi><mtext> </mtext></annotation-xml></semantics></math>are real numbers.
two-sided limit
the limit of a function<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></annotation-xml></semantics></math>as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a,is equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> L,that is,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→a f(x)=L  if and only if<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ a − f(x)= lim x→ a + f(x).