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Mathematics LibreTexts

12-2. Finding Limits: Properties of Limits

Finding Limits: Properties of Limits
In this section, you will:
  • Find the limit of a sum, a difference, and a product.
  • Find the limit of a polynomial.
  • Find the limit of a power or a root.
  • Find the limit of a quotient.

Consider the rational 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

The function can be factored as follows:

<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−7 ) ( x+1 ) x−7 , which gives us f(x)=x+1,x≠7.

Does this mean 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>is the same as 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> g(x)=x+1?

The answer is no. 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>does not have<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  in its domain, but<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>g</mi><mtext> </mtext></annotation-xml></semantics></math>does. Graphically, we observe there is a hole in 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 )  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, as shown in [link] and no such hole in 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> g( x ), as shown in [link].

<figure class="small" id="CNX_Precalc_Figure_12_02_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>The graph of 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>contains a break at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>7</mn></mrow></annotation-xml></semantics></math> and is therefore not continuous 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 where f(x) = (x^2-6x-7)\(x-7) with a discontinuity at (7, 8)</figure> <figure class="small" id="CNX_Precalc_Figure_12_02_002" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>The graph of function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>g</mi><mtext> </mtext></annotation-xml></semantics></math>is continuous.</figcaption> Graph of an increasing function where g(x) = x+1</figure>

So, do these two different functions also have different limits 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?

Not necessarily. Remember, in determining a limit of a 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, what matters is whether the output approaches a real number as we 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> x=a.  The existence of a limit does not depend on what happens 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>equals<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.

Look again at [link] and [link]. Notice that in both graphs, 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, the output values approach 8. This means

<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)= lim x→7 g(x).

Remember that when determining a limit, the concern is what occurs 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, 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=a.  In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.

Finding the Limit of a Sum, a Difference, and a Product

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

Properties of Limits

Let<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, k, A, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>B</mi><mtext> </mtext></annotation-xml></semantics></math>represent real numbers, and<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>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>g</mi><mtext> </mtext></annotation-xml></semantics></math>be functions, 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> lim x→a f(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> lim x→a g(x)=B.  For limits that exist and are finite, the properties of limits are summarized in [link]

Constant, k <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→a k=k
Constant times a function <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→a [ k⋅f(x) ]=k lim x→a f(x)=kA
Sum of functions <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→a [ f(x)+g(x) ]= lim x→a f(x)+ lim x→a g(x)=A+B
Difference of functions <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→a [ f(x)−g(x) ]= lim x→a f(x)− lim x→a g(x)=A−B
Product of functions <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→a [ f(x)⋅g(x) ]= lim x→a f(x)⋅ lim x→a g(x)=A⋅B
Quotient of functions <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→a f(x) g(x) = lim x→a f(x) lim x→a g(x) = A B ,B≠0
Function raised to an exponent <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→a [f(x)] n = [ lim x→∞ f(x) ] n = A n , where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is a positive integer
nth root of a function, where n is a positive integer <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→a f(x) n = lim x→a [ f(x) ] n = A n
Polynomial function <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→a p(x)=p(a)
Evaluating the Limit of a Function Algebraically

Evaluate<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→3 ( 2x+5 ).

<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><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→3 (2x+5)= lim x→3 (2x)+ lim x→3 (5) Sum of functions property                      = 2lim x→3 (x)+ lim x→3 (5)Constant times a function property                      =2(3)+5  Evaluate                      =11

Evaluate 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→−12 ( −2x+2 ).

26

Finding the Limit of a Polynomial

Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit of a polynomial 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><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>is equivalent to simply evaluating the function for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi></annotation-xml></semantics></math>.

Given a function containing a polynomial, find its limit.

Use the properties of limits to break up the polynomial into individual terms.
Find the limits of the individual terms.
Add the limits together.
Alternatively, evaluate the function for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi></annotation-xml></semantics></math>.

 

Evaluating the Limit of a Function Algebraically

Evaluate<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→3 ( 5 x 2 ).

<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><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→3 (5 x 2 )=5 lim x→3 ( x 2 ) Constant times a function property                 =5( 3 2 )Function raised to an exponent property                 =45

Evaluate<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→4 ( x 3 −5).

59

Evaluating the Limit of a Polynomial Algebraically

Evaluate<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→5 ( 2 x 3 −3x+1 ).

<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><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→5 (2 x 3 −3x+1)= lim x→5 (2 x 3 )− lim x→5 (3x)+ lim x→5 (1) Sum of functions                                = 2lim x→5 (x 3 )− 3lim x→5 (x)+ lim x→5 (1) Constant times a function                                =2( 5 3 )−3(5)+1Function raised to an exponent                                =236 Evaluate

Evaluate 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→−1 ( x 4 −4 x 3 +5 ).

10

Finding the Limit of a Power or a Root

When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.

Evaluating a Limit of a Power

Evaluate<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+1 ) 5 .

We will take the limit 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 2 and raise the result to the 5th power.

<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><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→2 (3x+1) 5 = ( lim x→2 (3x+1)) 5                       = (3(2)+1) 5                       = 7 5                       =16,807

Evaluate the following 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→−4 ( 10x+36 ) 3 .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>64</mn></mrow></annotation-xml></semantics></math>

If we can’t directly apply the properties of a limit, for example in <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><munder><mrow><mi>lim</mi></mrow></munder></annotation-xml></semantics></math> x→2 ( x 2 +6x+8 x−2 ), can we still determine the limit of the function as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>x</mi></annotation-xml></semantics></math> approaches <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>a</mi></annotation-xml></semantics></math>?

Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.

Finding the Limit of a Quotient

Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.

Given the limit of a function in quotient form, use factoring to evaluate it.

  1. Factor the numerator and denominator completely.
  2. Simplify by dividing any factors common to the numerator and denominator.
  3. Evaluate the resulting limit, remembering to use the correct domain.
Evaluating the Limit of a Quotient by Factoring

Evaluate<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 ( x 2 −6x+8 x−2 ).

Factor where possible, and simplify.

<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><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→2 ( x 2 −6x+8 x−2 )= lim x→2 ( (x−2)(x−4) x−2 ) Factor the numerator.                               = lim x→2 ( (x−2) (x−4)x−2 ) Cancel the common factors.                               = lim x→2 (x−4) Evaluate.                               =2−4=−2
Analysis

When the limit of a rational function cannot be evaluated directly, factored forms of the numerator and denominator may simplify to a result that can be evaluated.

Notice, 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+8 x−2

is equivalent to 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><mi>x</mi><mo>−</mo><mn>4</mn><mo>,</mo><mi>x</mi><mo>≠</mo><mn>2.</mn></mrow></annotation-xml></semantics></math>

Notice that the limit exists even though the function 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 = 2.

Evaluate 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→7 ( x 2 −11x+28 7−x ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>3</mn></mrow></annotation-xml></semantics></math>

Evaluating the Limit of a Quotient by Finding the LCD

Evaluate<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→5 ( 1 x − 1 5 x−5 ).

Find the LCD for the denominators of the two terms in the numerator, and convert both fractions to have the LCD as their denominator.

Analysis

When determining the limit of a rational function that has terms added or subtracted in either the numerator or denominator, the first step is to find the common denominator of the added or subtracted terms; then, convert both terms to have that denominator, or simplify the rational function by multiplying numerator and denominator by the least common denominator. Then check to see if the resulting numerator and denominator have any common factors.

Evaluate<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→−5 ( 1 5 + 1 x 10+2x ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 50

Given a limit of a function containing a root, use a conjugate to evaluate.

  1. If the quotient as given is not in indeterminate<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 0 )  form, evaluate directly.
  2. Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the least common denominator (LCD).
  3. If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the conjugateof the numerator. 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> a± b  are conjugates.
  4. Simplify.
  5. Evaluate the resulting limit.
Evaluating a Limit Containing a Root Using a Conjugate

Evaluate<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 ( 25−x −5 x ).

<math display="inline" 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><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→0 ( 25−x −5 x )= lim x→0 ( ( 25−x −5 ) x ⋅ ( 25−x +5 ) ( 25−x +5 ) )Multiply numerator and denominator by the conjugate.                                = lim x→0 ( ( 25−x )−25 x( 25−x +5 ) )Multiply: ( 25−x −5 )⋅( 25−x +5 )=( 25−x )−25.                                = lim x→0 ( −x x( 25−x +5 ) ) Combine like terms.                               = lim x→0 ( − x x ( 25−x +5 ) ) Simplify  −x x =−1.                                = −1 25−0 +5 Evaluate.                               = −1 5+5 =− 1 10

Analysis

When determining a limit of a function with a root as one of two terms where we cannot evaluate directly, think about multiplying the numerator and denominator by the conjugate of the terms.

Evaluate 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 h→0 ( 16−h −4 h ).

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

Evaluating the Limit of a Quotient of a Function by Factoring

Evaluate<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→4 ( 4−x x −2 ).

<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><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→4 ( 4−x x −2 ) =  lim x→4 ( (2+ x )(2− x ) x −2 ) Factor.                       = lim x→4 ( (2+ x ) (2− x ) − (2− x ) )Factor −1 out of the denominator. Simplify.                       = lim x→4 −(2+ x ) Evaluate.                       =−(2+ 4 )                      =−4

Analysis

Multiplying by a conjugate would expand the numerator; look instead for factors in the numerator. Four is a perfect square so that the numerator is in the form

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>a</mi></msup></mrow></annotation-xml></semantics></math> 2 − b 2

and may be factored as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> a+b )( a−b ).

Evaluate 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→3 ( x−3 x − 3 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msqrt/></mrow></annotation-xml></semantics></math> 3

Given a quotient with absolute values, evaluate its limit.

  1. Try factoring or finding the LCD.
  2. If the limit cannot be found, choose several values close to and on either side of the input where the function is undefined.
  3. Use the numeric evidence to estimate the limits on both sides.
Evaluating the Limit of a Quotient with Absolute Values

Evaluate<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→7 | x−7 | x−7 .

The function is undefined 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 ,so we will try values close to 7 from the left and the right.

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> | 6.9−7 | 6.9−7 = | 6.99−7 | 6.99−7 = | 6.999−7 | 6.999−7 =−1

Right-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> | 7.1−7 | 7.1−7 = | 7.01−7 | 7.01−7 = | 7.001−7 | 7.001−7 =1

Since the left- and right-hand limits are not equal, there is no limit.

Evaluate<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→ 6 + 6−x | x−6 | .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−1</mn></mrow></annotation-xml></semantics></math>

Access the following online resource for additional instruction and practice with properties of limits.

Key Concepts

  • The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See [link].
  • The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See [link] and[link].
  • The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See [link].
  • The limit of the root of a function equals the corresponding root of the limit of the function.
  • One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See [link].
  • Another method of finding the limit of a complex fraction is to find the LCD. See [link].
  • A limit containing a function containing a root may be evaluated using a conjugate. See [link].
  • The limits of some functions expressed as quotients can be found by factoring. See [link].
  • One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See [link].

Section Exercises

Verbal

Give an example of a type of 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>whose 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, 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 ).

If<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>is a polynomial function, the limit of a polynomial 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><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>will always 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> f( a ).

When direct substitution is used to evaluate the limit of a rational 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><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and the result 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 )= 0 0 ,does this mean that the limit of<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>does not exist?

What does it mean to say 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 ) , 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>c</mi><mo>,</mo></annotation-xml></semantics></math> is undefined?

It could mean either (1) the values of the function increase or 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<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, or (2) the left and right-hand limits are not equal.

Algebraic

For the following exercises, evaluate the limits algebraically.

<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 ( 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 ( −5x x 2 −1 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>−</mo><mn>10</mn></mrow></mfrac></mrow></annotation-xml></semantics></math> 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 ( x 2 −5x+6 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→3 ( x 2 −9 x−3 )

6

<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 −2x−3 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→ 3 2 ( 6 x 2 −17x+12 2x−3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 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→− 7 2 ( 8 x 2 +18x−35 2x+7 )

<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 −9 x−5x+6 )

6

<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 ( −7 x 4 −21 x 3 −12 x 4 +108 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→3 ( x 2 +2x−3 x−3 )

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> h→0 ( ( 3+h ) 3 −27 h )

<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> h→0 ( ( 2−h ) 3 −8 h )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>12</mn></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> h→0 ( ( h+3 ) 2 −9 h )

<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> h→0 ( 5−h − 5 h )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 10

<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 ( 3−x − 3 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→9 ( x 2 −81 3− x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>108</mn></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 ( x − x 2 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 ( x 1+2x −1 )

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 2 ( x 2 − 1 4 2x−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→4 ( x 3 −64 x 2 −16 )

6

<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 − ( |x−2| 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 + ( | x−2 | x−2 )

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 ( | x−2 | 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→ 4 − ( | x−4 | 4−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→ 4 + ( | x−4 | 4−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 ( | x−4 | 4−x )

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→2 ( −8+6x− x 2 x−2 )

For the following exercise, use the given information to evaluate the limits:<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→c f(x)=3,<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→c g( x )=5

<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→c  [  2f(x)+ g(x)   ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><mo>+</mo><msqrt/></mrow></annotation-xml></semantics></math> 5

<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→c  [  3f(x)+ g(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→c f(x) g(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics></math> 5

For the following exercises, evaluate the following limits.

<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 cos( πx )

<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 sin( πx )

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→2 sin( π x )

<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><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 2 x 2 +2x+1, x≤0 x−3,  x>0 ;  lim x→ 0 + f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>3</mn></mrow></annotation-xml></semantics></math>

<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><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 2 x 2 +2x+1, x≤0 x−3,  x>0 ;  lim x→ 0 − f(x)

<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><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 2 x 2 +2x+1, x≤0 x−3,  x>0 ;  lim x→0 f(x)

does not exist; right-hand limit is not the same as the left-hand 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→4 x+5 −3 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 + (2x−〚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 x+7 −3 x 2 −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→ 3 + x 2 x 2 −9

Limit does not exist; limit approaches infinity.

For the following exercises, find the average rate of change<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+h)−f(x) h .

<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><mi>x</mi><mo>+</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

<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><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>h</mi></mrow></annotation-xml></semantics></math>

<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> x 2 +3x+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><msup/></mrow></annotation-xml></semantics></math> x 2 +4x−100

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>h</mi><mo>+</mo><mn>4</mn></mrow></annotation-xml></semantics></math>

<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><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +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><mi>cos</mi><mo stretchy="false">(</mo><mi>x</mi><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><mfrac><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></annotation-xml></semantics></math> h

<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><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 3 −4x

<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

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

<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

<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><msqrt/></mrow></annotation-xml></semantics></math> x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></annotation-xml></semantics></math> x+h + x

Graphical

Find an equation that could be represented by [link].

<figure class="small" id="CNX_Precalc_Figure_12_02_201">Graph of increasing function with a removable discontinuity at (2, 3).</figure>

Find an equation that could be represented by [link].

<figure class="small" id="CNX_Precalc_Figure_12_02_202">Graph of increasing function with a removable discontinuity at (-3, -1).</figure>

<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> x )= x 2 +5x+6 x+3

For the following exercises, refer to [link].

<figure class="small" id="CNX_Precalc_Figure_12_02_203">Graph of increasing function from zero to positive infinity.</figure>

What is the right-hand limit 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 0?

What is the left-hand limit 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 0?

does not exist

Real-World Applications

The position 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> s(t)=−16 t 2 +144t  gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on 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> [ 1,2 ].

The height of a projectile is given 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> s(t)=−64 t 2 +192t  Find the average rate of change of the height from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=1  second 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> t=1.5  seconds.

52

The amount of money in an account after<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>years compounded continuously at 4.25% interest is given by the formula<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= A 0 e 0.0425t ,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> A 0  is the initial amount invested. Find the average rate of change of the balance of the account from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=1  year 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> t=2  years if the initial amount invested is $1,000.00.

Glossary

properties of limits
a collection of theorems for finding limits of functions by performing mathematical operations on the limits