2.2: The Limit of a Function - A Numerical and Graphical Investigation

Note to the Instructor (click to expand) This section should be used to introduce the intuitive idea of a limit at a finite number by way of building tables and using non-graphing technology (graphs are saved for the next section). Note that limits at infinity are avoided at this point because we are using computational technology to approximate limit values. Single-sided limits are introduced (again, intuitively) and the "DNE" notation is presented. Warnings should abound that this is the weakest form of limit approximation. Core Prerequisites (i.e., "Understanding the Lesson"): The following prerequisite topics (from your past math courses) are required to understand the core concepts in this lesson. (click to expand) An Overview of Functions The Definition of a Function Graphing Functions Piecewise Functions  Trigonometry Converting Between Degrees and Radians Operational Prerequisites (i.e., "Completing the Homework"): The following prerequisite topics are needed to complete the homework. (click to expand) Arithmetic The Order of Operations The Absolute Value of a Number Simplifying Expressions Algebraic Vocabulary and Evaluating Expressions An Overview of Functions Evaluating Functions Trigonometry Evaluating Trigonometric Functions Using Technology The following is a list of learning objectives for this section. Learning Objectives (click to expand) Describe the limit of a function and use proper limit notation. Use a table of values to estimate the limit of a function or to identify when the limit does not exist. Define and investigate one-sided limits. Explain the relationship between one-sided and two-sided limits. Using correct notation, describe an infinite limit. Explain why investigating limits using technology is inherently flawed.

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The concept of a limit (or limiting process) has been around for thousands of years. Early mathematicians used limits to obtain more accurate approximations of areas of circles. Yet, the formal definition of a limit  - as we know and understand it today - did not appear until the late 19^th century. Therefore, we begin our quest to understand limits, as our mathematical ancestors did, using an intuitive approach. Only after we have had time to play with this intuitive approach do we examine the formal definition of a limit.

Finite Limit at a Finite Number

Let's consider how the function \(f(x)= \frac{x^2−4}{x−2}\) behaves around \(x=2\). Building a quick table of values, we get the following.

Table \(\PageIndex{1}\): Investigating the behavior of \(f(x)= \frac{x^2−4}{x−2}\) near \(x=2\) 

\(x\)	\(f(x)\)	\(x\)	\(f(x)\)
\(1.5\)	3.5	\(2.5\)	4.5
\(1.9\)	3.9	\(2.1\)	4.1
\(1.99\)	3.99	\(2.01\)	4.01
\(1.999\)	3.999	\(2.001\)	4.001

As the values of \(x\) approach \(2\) (from both sides of \(2\)), the values of \(y=f(x)\) seem to be approaching \(4\). If correct, we would then say that the limit of \(f(x)\) as \(x\) approaches \(2\) is \(4\). Symbolically, we express this limit as\[ \lim_{x \to 2} \dfrac{x^2 - 4}{x - 2} =4.\nonumber \]From this very brief informal look at one limit, let's start to develop an intuitive definition of the finite limit at a finite number.

We can think of the limit of a function at a finite number \(a\) as being the single finite number \(L\) (the finite limit) that the function values approach as the \(x\)-values approach \(a\), provided such a real number \(L\) exists. Stated more carefully, we have the following intuitive, but not precise, definition.

Definition (Intuitive): Finite Limit at a Finite Number

Let \(f(x)\) be a function defined at all values in an open interval containing a finite number \(a\), with the possible exception of \(a\) itself, and let \(L\) be a finite real number. If the values of the function \(f(x)\) approach the real number \(L\) as the values of \(x\) (\(\neq a\)) approach the number \(a\), then we say that the finite limit of \(f(x)\) as \(x\) approaches the finite number \(a\) is \(L\). (Said more succinctly, as \(x\) gets closer to \(a\), \(f(x)\) gets closer to \(L\).) Symbolically, we express this idea as\[\lim_{x \to a} f(x)=L. \nonumber \]Often, this is simply referred to as a limit (rather than a finite limit at a finite number).

Understanding the concept of a limit is so important in Calculus that we will spend quite some time getting comfortable with it before finally (several sections from now) going deep into the precise definition of a limit.

Caution: \(x \neq a\)

The intuitive definition of a finite limit at a finite number states that \(x\) need not equal \(a\). This is critical to understand! The \(x\)-values are allowed to get very close to \(a\) but will never become \(a\). In a very real sense, the limit,\[ \lim_{x \to a} f(x)=L, \nonumber \]is concerned with the function values as \(x\) gets closer and closer to \(a\); however, it is not asking, "What is \(f(a)\)?" In fact, it could be the case that \(f(a)\) does not exist or \(L \neq f(a)\).

In this section, we explore how to estimate the value of a limit using a combination of tables and computational technology. We will refer to this method of investigation as a numerical approximation of the limit. This can also be rephrased as, "investigating the limit numerically." In practice, using a numerical approximation to state the value of a limit presents three issues:

1. it can lead to incorrect presumptions about the true value of the limit,
2. as Calculus is the study of the infinite and infinitesimal, and technology uses finite precision arithmetic, limit computations attempted via technology can lead to massively incorrect values, and
3. using technology or tables to gain an approximation of the value of a limit is not Calculus - it is button pushing. Your task in Calculus is to gain a demonstrable understanding of the fundamental concepts of Calculus, and this includes how to work with limits without touching technology or building tables.

These warnings stated, we still take the time to investigate limits numerically in order to gain a stronger intuitive understanding of limits.

Using Tables to Estimate Limits

To begin our investigation of evaluating \( \displaystyle \lim_{x \to a} f(x) \) numerically, we (again) build a table of function values. Choosing two sets of \(x\)-values - one set approaching \(a\) from "below" (these are \( x \)-values less than \(a\)), and another set of values approaching \(a\) from "above" (these are \( x \)-values greater than \(a\)). Doing so yields something like Table \(\PageIndex{2}\).

Table \(\PageIndex{2}\): Typical table investigating the behavior of \(f\) near \(x = a\)

\(x\)	\(f(x)\)	\(x\)	\(f(x)\)
\(a-0.1\)	\(f(a-0.1)\)	\(a+0.1\)	\(f(a+0.1)\)
\(a-0.01\)	\(f(a-0.01)\)	\(a+0.01\)	\(f(a+0.01)\)
\(a-0.001\)	\(f(a-0.001)\)	\(a+0.001\)	\(f(a+0.001)\)
\(a-0.0001\)	\(f(a-0.0001)\)	\(a+0.0001\)	\(f(a+0.0001)\)
Use additional values as necessary.		Use additional values as necessary.

Next, we look at the values in each of the \(f(x)\) columns and determine whether the values seem to be approaching a single value as we move down the column. In our columns, we look at the sequence of function values \(f(a−0.1)\), \(f(a−0.01)\), \(f(a−0.001)\), \(f(a−0.0001)\), and so on, and the sequence of function values \(f(a+0.1)\), \(f(a+0.01)\), \(f(a+0.001)\), \(f(a+0.0001)\), and so on.

Caution: Be prepared to try "wiggle" sizes other than powers of ten

We have chosen the \(x\)-values \(a \pm 0.1, \, a \pm 0.01, \, a \pm 0.001, \, a \pm 0.0001\), and so forth. These "wiggle" sizes (\(\pm 0.1\), \(\pm 0.01\), \(\pm 0.001\), and \( \pm 0.0001 \)) will work most of the time; however, there will be occasions where we might need to modify our choice of "wiggle" sizes to something less predictable.

If both function value columns approach a common, \(L\), we are inclined to think that\[\lim_{x \to a}f(x)=L.\nonumber \]If unsure of our result, we could evaluate the given function at values closer (but not equal) to \( a \); however, due to finite precision arithmetic, the technology we use will round our input to \( a \) if our "wiggle" from \( a \) is too small.

For example, when we were trying to evaluate \( \displaystyle \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \) using a table of values, Desmos returns Table \(\PageIndex{3}\).

Table \(\PageIndex{3}\): Showcasing how technology can fail us

\(x\)	\(f(x) = \frac{x^2 - 4}{x - 2}\)	\(x\)	\(f(x) = \frac{x^2 - 4}{x - 2}\)
\(1.9\)	\(3.9\)	\(2.1\)	\(4.1\)
\( 1.9999 \)	\( 3.9999 \)	\( 2.0001 \)	\( 4.0001 \)
\(1.9999999\)	\(3.9999999\)	\(2.0000001\)	\(4.0000001\)
\(1.9999999999999999\)	undefined	\(2.0000000000000001\)	undefined

In this case, the "wiggle" sizes we are choosing are too small for Desmos to handle.

It's time we practice some of these new skills.

Example \(\PageIndex{1}\)

Evaluate\[\lim_{\theta \to 0}\frac{\sin \theta}{\theta}\nonumber \]using a table of function values.

Solution

We have calculated some values of \(f(\theta)=\frac{\sin \theta}{\theta}\) for \(\theta\) near to \(0\) in Table \(\PageIndex{4}\).

Table \(\PageIndex{4}\): Investigating the values of \( \frac{\sin(\theta)}{\theta} \) as \(\theta\) approaches \(0\)

\(\theta\)	\(\frac{\sin \theta}{\theta}\)	\(\theta\)	\(\frac{\sin \theta}{\theta}\)
-0.1	0.99833417	0.1	0.99833417
-0.01	0.99998333	0.01	0.99998333
-0.001	0.99999983	0.001	0.99999983
-0.0001	1	0.0001	1

As we read down each \(\frac{\sin \theta}{\theta}\) column, we see that the values appear to be approaching \( 1 \). In fact, Desmos (the technology I used to build this table of values) is confidently stating that the function evaluated at \( \theta = \pm 0.001 \) is exactly 1; however, this is not true. In fact, the value of \( \frac{\sin ( \pm0.001)}{\pm 0.001} \) is approximately 0.9999998333. Nevertheless, it is somewhat reasonable to conclude that\[\lim_{\theta \to 0}\dfrac{\sin \theta}{\theta}=1 \nonumber \]based on this table of values.

Before moving to another example, it's time I caution you about trigonometric functions and Calculus. 

Caution: Calculus Does Not Work in Degrees

For reasons we will discuss later, when working with trigonometric functions in Calculus, you must work in radians - not degrees.

Example \(\PageIndex{2}\)

Evaluate\[\lim_{x \to 4}\frac{\sqrt{x}−2}{x−4}\nonumber \]using a table of function values.

Solution

As before, we use a table to list values of the function as \(x\) gets close to \( 4 \) (from both sides of \( 4 \)).

Table \(\PageIndex{5}\): Investigating the behavior of \(\frac{\sqrt{x}−2}{x−4}\) as \(x \to 4\)

\(x\)	\(\frac{\sqrt{x}−2}{x−4}\)	\(x\)	\(\frac{\sqrt{x}−2}{x−4}\)
3.9	0.251582341869	4.1	0.248456731317
3.99	0.25015644562	4.01	0.24984394501
3.999	0.250015627	4.001	0.249984377
3.9999	0.250001563	4.0001	0.249998438
3.99999	0.25000016	4.00001	0.24999984

After inspecting this table, the function values for \( x \lt 4 \) appear to decrease toward \(0.25\). Likewise, the function values for \( x \gt 4 \) appear to be increasing toward \(0.25\). We conclude that\[\lim_{x\to4}\dfrac{\sqrt{x}−2}{x−4}=0.25.\nonumber \]

Checkpoint \(\PageIndex{2}\)

Two Important Limits

Generating a table of function values provides useful insight into the value of a function's limit at a given point. However, this technique relies too much on technology and the hope that your technology is not "lying" to you. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature, and we explore them later in this chapter. At this point, we introduce two particular limits foundational to the techniques to come.

Theorem: Two Important Limits

Let \(a\) be a real number and \(c\) be a constant.

1. \(\displaystyle \lim_{x \to a}x=a\) (Limit of the Identity Function)
2. \(\displaystyle \lim_{x \to a}c=c\) (Limit of a Constant)

We can make the following observations about these two limits.

1. For the first limit, observe that as \(x\) approaches \(a\), so does \(f(x)\), because \(f(x)=x\). Consequently, \(\displaystyle \lim_{x \to a}x=a\).
2. For the second limit, consider Table \(\PageIndex{6}\).

Table \(\PageIndex{6}\): Investigating what a constant does as we change \(x\)

\(x\)	\(f(x)=c\)	\(x\)	\(f(x)=c\)
\(a-0.1\)	\(c\)	\(a+0.1\)	\(c\)
\(a-0.01\)	\(c\)	\(a+0.01\)	\(c\)
\(a-0.001\)	\(c\)	\(a+0.001\)	\(c\)
\(a-0.0001\)	\(c\)	\(a+0.0001\)	\(c\)

Observe that for all values of \(x\), the value of \(f(x)\) remains constant. We have no choice but to conclude \(\displaystyle \lim_{x \to a}c=c\).

Caution: Observations Are Not Proofs

As you move through this course (and toward higher-level courses in Mathematics), you must have a clear understanding between an observation and a proof. What we have observed above (that \( \displaystyle \lim_{x \to a} c \) looks to be \( c \)) is not a proof. We will learn how to prove these types of statements later.

The Existence of a Limit

As we consider the limit in the following example, keep in mind that for a function's limit to exist at a point, the function values must approach a single real number. If the function values do not approach a single value, then the limit does not exist.

Example \(\PageIndex{3}\)

Evaluate\[\lim_{x \to 0}\sin\left(\dfrac{1}{x}\right)\nonumber \]using a table of values.

Solution

Table \(\PageIndex{7}\) lists values for the function \(\sin(1/x)\) as \( x \) approaches \( 0 \) from both the left and the right.

Table \(\PageIndex{7}\): Investigating the behavior of \(\sin\left(\frac{1}{x}\right)\) as \(x \to 0\)

\(x\)	\(\sin(1/x)\)	\(x\)	\(\sin(1/x)\)
-0.1	0.544021110889	0.1	−0.544021110889
-0.01	0.50636564111	0.01	−0.50636564111
-0.001	−0.8268795405312	0.001	0.8268795405312
-0.0001	0.305614388888	0.0001	−0.305614388888
-0.00001	−0.035748797987	0.00001	0.035748797987
-0.000001	0.349993504187	0.000001	−0.349993504187

After examining the table of function values, we can see that the \(y\)-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let's take a more systematic approach. Take the following sequence of \(x\)-values approaching \(0\):\[\dfrac{2}{ \pi },\, \dfrac{2}{3 \pi },\, \dfrac{2}{5 \pi },\, \dfrac{2}{7 \pi },\, \dfrac{2}{9 \pi },\, \dfrac{2}{11 \pi },\, \ldots .\nonumber \]The corresponding \(y\)-values are\[1,\, -1,\, 1,\, -1,\, 1,\, -1,\, \ldots. \nonumber \]At this point, we conclude that\[\lim_{x \to 0} \sin\left( \dfrac{1}{x} \right)\nonumber \]does not exist. Mathematicians frequently abbreviate "does not exist" as DNE. Thus, we would write\[\lim_{x \to 0} \sin\left( \dfrac{1}{x} \right) \quad \text{DNE}.\nonumber \]

Notation: DNE

When a limit does not become a single value (whether finite or infinity), it is an agreed mathematical convention to write DNE, which tells the reader the limit does not exist.

Checkpoint \(\PageIndex{3}\)

One-Sided Limits

Sometimes, indicating that the limit of a function fails to exist at a number does not provide us with enough information about the behavior of the function near that particular number. To see this, consider the function\[g(x)= \dfrac{|x−2|}{x−2}.\nonumber \]

Table \(\PageIndex{8}\): Investigating the behavior of \(\frac{|x−2|}{x−2}\) as \(x \to 2\)

\(x\)	\(g(x) = \frac{|x - 2|}{x - 2}\)	\(x\)	\(g(x) = \frac{|x - 2|}{x - 2}\)
\(1.9\)	\(-1\)	\(2.1\)	\(1\)
\( 1.99 \)	\( -1 \)	\( 2.01 \)	\( 1 \)
\(1.999\)	\(-1\)	\(2.001\)	\(1\)
\(1.9999\)	\( -1 \)	\(2.0001\)	\( 1 \)

As we pick values of \(x\) close to \(2\), \(g(x)\) does not approach a single value, so the limit as \(x\) approaches \(2\) does not exist. That is,\[\lim_{x \to 2}g(x) \, \, \text{DNE}.\nonumber \]However, this statement alone does not give us a complete picture of the behavior of the function around \(x = 2\). To provide a more accurate description, we introduce the idea of a one-sided limit (also known as a single-sided limit).

For all values to the left of \(2\) (or the "below" \(2\)), \(g(x)=−1\). Thus, as \(x\) approaches \(2\) from the left, \(g(x)\) approaches \(−1\). Mathematically, we say that the limit as \(x\) approaches \(2\) from the left is \(−1\). Symbolically, we express this idea as\[\lim_{x \to 2^−}g(x)=−1. \nonumber \]Similarly, as \(x\) approaches \(2\) from the right (or from "above"), \(g(x)\) approaches \(1\). Symbolically, we express this idea as\[\lim_{x \to 2^+}g(x)=1.\nonumber \]We can now present an informal definition of one-sided limits.

Definition: One-sided Limits

Limit from the left: Let \(f(x)\) be a function defined at all values in an open interval of the form \((z,a)\), and let \(L\) be a real number. If the values of the function \(f(x)\) approach the real number \(L\) as the values of \(x\) (where \(x<a\)) approach the number \(a\), then we say that \(L\) is the limit of \(f(x)\) as \(x\) approaches \(a\) from the left. Symbolically, we express this idea as\[\lim_{x \to a^−}f(x)=L. \nonumber \]

Limit from the right:  Let \(f(x)\) be a function defined at all values in an open interval of the form \((a,c)\), and let \(L\) be a real number. If the values of the function \(f(x)\) approach the real number \(L\) as the values of \(x\) (where \(x>a\)) approach the number \(a\), then we say that \(L\) is the limit of \(f(x)\) as \(x\) approaches \(a\) from the right. Symbolically, we express this idea as\[\lim_{x \to a^+}f(x)=L. \nonumber \]

Example \(\PageIndex{4}\)

For the function\[f(x)=\begin{cases}x+1, & \text{if }x<2\\
x^2−4, & \text{if }x \geq 2
\end{cases}, \nonumber \]evaluate each of the following limits.

1. \(\displaystyle \lim_{x \to 2^−}f(x)\)
2. \(\displaystyle \lim_{x \to 2^+}f(x)\)

Solutions

We can use tables of values again. Observe in Table \(\PageIndex{9}\), for values of \(x\) less than \(2\), we use \(f(x)=x+1\), and for values of \(x\) greater than \(2\), we use \(f(x)=x^2−4.\)

Table \(\PageIndex{9}\): Investigating the one-sided behavior of a piecewise-defined function

\(x\)	\(f(x)=x+1\)	\(x\)	\(f(x)=x^2-4\)
1.9	2.9	2.1	0.41
1.99	2.99	2.01	0.0401
1.999	2.999	2.001	0.004001
1.9999	2.9999	2.0001	0.00040001
1.99999	2.99999	2.00001	0.0000400001

Based on this table, we can conclude the following.\[ \lim_{x \to 2^−}f(x)=3 \quad \text{and} \quad \lim_{x \to 2^+}f(x)=0. \nonumber \]Therefore, the (two-sided) limit of \(f(x)\) does not exist at \(x=2\).

Checkpoint \(\PageIndex{4}\)

Let's now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. If the limit from the right and the limit from the left share a common value, then that is the limit of the function at that point. Similarly, suppose the limit from the left and the right take on different values. In that case, the limit of the function does not exist. These conclusions are summarized in the following theorem.

Theorem: Relating One-Sided and Two-Sided Limits

Let \(f(x)\) be a function defined at all values in an open interval containing \(a\), with the possible exception of \(a\) itself, and let \(L\) be a real number, \(\infty\), or \(-\infty\). Then,\[\lim_{x \to a}f(x)=L \nonumber \]if and only if\[\lim_{x \to a^−}f(x)=L \quad \text{and} \quad \lim_{x \to a^+} f(x)=L.\nonumber \]

Infinite Limit at a Finite Number

Evaluating the limit of a function at a point, or evaluating the limit of a function from the right and left at a point, helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

Consider the function \(h(x)= \dfrac{1}{(x−2)^2} \). Let's build a table of values to help us determine the value of \( \displaystyle \lim_{x \to 2} h(x) \).

Table \(\PageIndex{10}\): Investigating the values of \(\frac{1}{(x−2)^2} \) as \(x \to 2\)

\(x\)	\(h(x) = \frac{1}{(x - 2)^2}\)	\(x\)	\(h(x) = \frac{1}{(x - 2)^2}\)
\(1.9\)	\(100\)	\(2.1\)	\(100\)
\( 1.99 \)	\( 10000 \)	\( 2.01 \)	\( 10000 \)
\(1.999\)	\(1000000\)	\(2.001\)	\(1000000\)
\(1.9999\)	\( 10^8 \)	\(2.0001\)	\( 10^8 \)

We can clearly see that as the values of \(x\) approach \(2\), the values of \(h(x)= \frac{1}{(x−2)^2}\) become larger and larger (in fact, the values approach infinity). Mathematically, we say that the limit of \(h(x)\) as \(x\) approaches \(2\) is positive infinity. Symbolically, we express this idea as\[\lim_{x \to 2}h(x)=+\infty. \nonumber \]More generally, we define (again, in an intuitive, but not precise way) an infinite limit at a finite number as follows.

Definitions (Intuitive): Infinite Limits at Finite Numbers

Infinite limits from the left: Let \(f(x)\) be a function defined at all values in an open interval of the form \((b,a)\).

1. If the values of \(f(x)\) increase without bound as the values of \(x\) (where \(x<a\)) approach the number \(a\), then we say that the limit as \(x\) approaches \(a\) from the left is positive infinity and we write\[\lim_{x \to a^−}f(x)=+\infty. \nonumber \]
2. If the values of \(f(x)\) decrease without bound as the values of \(x\) (where \(x<a\)) approach the number \(a\), then we say that the limit as \(x\) approaches \(a\) from the left is negative infinity and we write\[\lim_{x \to a^−}f(x)=−\infty. \nonumber \]

Infinite limits from the right: Let \(f(x)\) be a function defined at all values in an open interval of the form \((a,c)\).

1. If the values of \(f(x)\) increase without bound as the values of \(x\) (where \(x>a\)) approach the number \(a\), then we say that the limit as \(x\) approaches \(a\) from the right is positive infinity and we write \[\lim_{x \to a^+}f(x)=+\infty. \nonumber \]
2. If the values of \(f(x)\) decrease without bound as the values of \(x\) (where \(x>a\)) approach the number \(a\), then we say that the limit as \(x\) approaches \(a\) from the right is negative infinity and we write \[\lim_{x \to a^+}f(x)=−\infty. \nonumber \]

Two-sided infinite limit: Let \(f(x)\) be defined for all \(x \neq a\) in an open interval containing \(a\)

1. If the values of \(f(x)\) increase without bound as the values of \(x\) (where \(x \neq a\)) approach the number \(a\), then we say that the limit as \(x\) approaches \(a\) is positive infinity and we write \[\lim_{x \to a} f(x)=+\infty. \nonumber \]
2. If the values of \(f(x)\) decrease without bound as the values of \(x\) (where \(x \neq a\)) approach the number \(a\), then we say that the limit as \(x\) approaches \(a\) is negative infinity and we write \[\lim_{x \to a}f(x)=−\infty. \nonumber \]

It is important to understand that when we write statements such as \(\displaystyle \lim_{x \to a}f(x)=\infty\) or \(\displaystyle \lim_{x \to a}f(x)=−\infty\), we are describing the behavior of the function. We are not asserting that the limit exists. Recall that for the limit of a function \(f\) to exist at \(a\), the function values must approach a finite, real number \(L\) as \(x\) approaches \(a\). That said, if \(\displaystyle \lim_{x \to a}f(x)=\infty\), we always write \(\displaystyle \lim_{x \to a}f(x)=\infty\) rather than saying the limit does not exist.¹

Example \(\PageIndex{5}\)

Evaluate each of the following limits, if possible.

1. \(\displaystyle \lim_{x \to 0^−} \frac{1}{x}\)
2. \(\displaystyle \lim_{x \to 0^+} \frac{1}{x}\)
3. \( \displaystyle \lim_{x \to 0}\frac{1}{x}\)

Solutions

Begin by constructing a table of function values.

Table \(\PageIndex{11}\): Investigating the value of \(\frac{1}{x}\) as \(x \to 0\)

\(x\)	\(\dfrac{1}{x}\)	\(x\)	\(\dfrac{1}{x}\)
-0.1	-10	0.1	10
-0.01	-100	0.01	100
-0.001	-1000	0.001	1000
-0.0001	-10,000	0.0001	10,000
-0.00001	-100,000	0.00001	100,000
-0.000001	-1,000,000	0.000001	1,000,000

The values of \(1/x\) decrease without bound as \(x\) approaches \(0\) from the left. We conclude that\[\lim_{x \to 0^−}\frac{1}{x}=−\infty.\nonumber \] The values of \(1/x\) increase without bound as \(x\) approaches \(0\) from the right. We conclude that\[\lim_{x \to 0^+}\frac{1}{x}=+\infty. \nonumber \] Since \(\displaystyle \lim_{x \to 0^−}\frac{1}{x}=−\infty\) and \(\displaystyle \lim_{x \to 0^+}\frac{1}{x}=+\infty\) have different values, we conclude that\[\lim_{x \to 0}\frac{1}{x}\quad\text{DNE.} \nonumber \]

Checkpoint \(\PageIndex{5}\)

A Final Warning About Using Tables

Up to this point, the only way you know how to evaluate the limit of a function is by building a table of values to estimate the limit. Although you will be allowed to while doing the homework in this section, you will rarely (if ever) use a table of values to approximate the value of a limit as we move forward. Moreover, consider the following warning.

Caution: Using Tables for Limits is a Terrible Idea

Outside of the homework in this section, attempting to estimate a limit using a table of values is horrendously bad. Again, read the following sentence carefully: 

Beyond this section, and unless specifically asked to do so, you are to never use a table of values to estimate a limit.

To illustrate why this warning is so important, consider the following example.

Example \(\PageIndex{6}\)

Evaluate\[\lim_{x \to 0}\sin \left(\dfrac{\pi}{x}\right)\nonumber \]using a table of functional values.

Solution

We have calculated the values of \(f(x)=\sin{\left( \frac{\pi}{x} \right)} \) for the values of \(x\) listed in Table \(\PageIndex{12}\).

Table \(\PageIndex{12}\): Showcasing why you cannot trust tables of values (in general)

\(x\)	\(\sin \left( \frac{\pi}{x} \right)\)	\(x\)	\(\sin \left( \frac{\pi}{x} \right)\)	\(x\)	\(\sin \left( \frac{\pi}{x} \right)\)	\(x\)	\(\sin \left( \frac{\pi}{x} \right)\)
0.1	0	0.3	-0.8660254	0.6	-0.8660254	0.7	-0.97492791
0.01	0	0.03	-0.8660254	0.06	0.8660254	0.07	0.78183148
0.001	0	0.003	-0.8660254	0.006	0.8660254	0.007	0.43388374
0.0001	0	0.0003	-0.8660254	0.0006	0.8660254	0.0007	0.97492791

The values chosen for \(x\) in each column approach \(0\); however, having the last digit as \(1\) tempts us into thinking the limit of this function is \(0\). Having the final digit as \(3\), makes \(-0.8660254\) seem like the limiting value. With a \(6\) as the final digit, we think the limit is \(0.8660254\). Finally, the \(7\) as the final digit leaves us completely confused. Is there an actual value for the limit? We will have to wait until the next section to see...

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¹ Some Calculus texts blur this line by stating limits of the form \( \displaystyle \lim_{x \to a} f(x) = \infty \) exist, while limits of the form \( \displaystyle \lim_{x \to a} f(x) = L \) exist and are finite. This is more of a definitional issue. For this text, a function having an infinite limit at a finite point is considered not to have a limit that exists at that finite point; however, we still describe the behavior of the function using the notation \( \displaystyle \lim_{x \to a} f(x) = \infty \) or \( \displaystyle \lim_{x \to a} f(x) = -\infty \).