2.2 The Limit of a Function

Intuitive Definition of a Limit

Let’s first take a closer look at how the function $f\left(x\right)=(x^2-4)\text{/}(x-2)$ behaves around $x=2$ in Figure 2.12. As the values of x approach 2 from either side of 2, the values of $y=f\left(x\right)$ approach 4. Mathematically, we say that the limit of $f\left(x\right)$ as x approaches 2 is 4. Symbolically, we express this limit as

From this very brief informal look at one limit, let’s start to develop an intuitive definition of the limit. We can think of the limit of a function at a number a as being the one real number L that the functional values approach as the x-values approach a, provided such a real number L exists. Stated more carefully, we have the following definition:

We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.

We apply this Problem-Solving Strategy to compute a limit in Example 2.4.

Example 2.4

Evaluating a Limit Using a Table of Functional Values 1

Evaluate $\underset{x\to 0}{\text{lim}}\frac{\text{sin}x}{x}$ using a table of functional values.

Solution

We have calculated the values of $f\left(x\right)=(\text{sin}x)\text{/}x$ for the values of x listed in Table 2.2.

x	$\frac{\text{sin}x}{x}$		x	$\frac{\text{sin}x}{x}$
−0.1	0.998334166468		0.1	0.998334166468
−0.01	0.999983333417		0.01	0.999983333417
−0.001	0.999999833333		0.001	0.999999833333
−0.0001	0.999999998333		0.0001	0.999999998333

Table 2.2 Table of Functional Values for $\underset{x\to 0}{\text{lim}}\frac{\text{sin}x}{x}$

Note: The values in this table were obtained using a calculator and using all the places given in the calculator output.

As we read down each $\frac{\left(\text{sin}x\right)}{x}$ column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that $\underset{x\to 0}{\text{lim}}\frac{\text{sin}x}{x}=1.$ A calculator or computer-generated graph of $f\left(x\right)=\frac{\left(\text{sin}x\right)}{x}$ would be similar to that shown in Figure 2.13, and it confirms our estimate.

Figure 2.13 The graph of $f\left(x\right)=(\text{sin}x)\text{/}x$ confirms the estimate from Table 2.2.

Example 2.5

Evaluating a Limit Using a Table of Functional Values 2

Evaluate $\underset{x\to 4}{\text{lim}}\frac{\sqrt{x}-2}{x-4}$ using a table of functional values.

Solution

As before, we use a table—in this case, Table 2.3—to list the values of the function for the given values of x.

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

Table 2.3 Table of Functional Values for $\underset{x\to 4}{\text{lim}}\frac{\sqrt{x}-2}{x-4}$

After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that $\underset{x\to 4}{\text{lim}}\frac{\sqrt{x}-2}{x-4}=0.25.$ We confirm this estimate using the graph of $f\left(x\right)=\frac{\sqrt{x}-2}{x-4}$ shown in Figure 2.14.

Figure 2.14 The graph of $f\left(x\right)=\frac{\sqrt{x}-2}{x-4}$ confirms the estimate from Table 2.3.

At this point, we see from Example 2.4 and Example 2.5 that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values. In Example 2.6, we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.

Example 2.6

Evaluating a Limit Using a Graph

For $g\left(x\right)$ shown in Figure 2.15, evaluate $\underset{x\to \mathrm{-1}}{\text{lim}}g\left(x\right).$

Figure 2.15 The graph of $g\left(x\right)$ includes one value not on a smooth curve.

Solution

Despite the fact that $g\left(\mathrm{-1}\right)=4,$ as the x-values approach −1 from either side, the $g\left(x\right)$ values approach 3. Therefore, $\underset{x\to \mathrm{-1}}{\text{lim}}g\left(x\right)=3.$ Note that we can determine this limit without even knowing the algebraic expression of the function.

Based on Example 2.6, we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.

Checkpoint 2.5

Use the graph of $h\left(x\right)$ in Figure 2.16 to evaluate $\underset{x\to 2}{\text{lim}}h\left(x\right),$ if possible.

Figure 2.16

Looking at a table of functional values or looking at the graph of a function provides us with useful insight into the value of the limit of a function at a given point. However, these techniques rely too much on guesswork. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the next section; however, at this point we introduce two special limits that are foundational to the techniques to come.

We can make the following observations about these two limits.

1. For the first limit, observe that as x approaches a, so does $f\left(x\right),$ because $f\left(x\right)=x.$ Consequently, $\underset{x\to a}{\text{lim}}x=a.$
2. For the second limit, consider Table 2.4.

Observe that for all values of x (regardless of whether they are approaching a), the values $f\left(x\right)$ remain constant at c. We have no choice but to conclude $\underset{x\to a}{\text{lim}}c=c.$

The Existence of a Limit

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

Example 2.7

Evaluating a Limit That Fails to Exist

Evaluate $\underset{x\to 0}{\text{lim}}\text{sin}\left(1\text{/}\mathit{\text{x}}\right)$ using a table of values.

Solution

Table 2.5 lists values for the function $\text{sin}(1\text{/}x)$ for the given values of x.

x	$\text{sin}\left(\frac{1}{x}\right)$		x	$\text{sin}\left(\frac{1}{x}\right)$
−0.1	0.544021110889		0.1	−0.544021110889
−0.01	0.50636564111		0.01	−0.50636564111
−0.001	−0.8268795405312		0.001	0.826879540532
−0.0001	0.305614388888		0.0001	−0.305614388888
−0.00001	−0.035748797987		0.00001	0.035748797987
−0.000001	0.349993504187		0.000001	−0.349993504187

Table 2.5 Table of Functional Values for $\underset{x\to 0}{\text{lim}}\text{sin}\left(\frac{1}{x}\right)$

After examining the table of functional 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:

$$\frac{2}{\pi },\frac{2}{3\pi },\frac{2}{5\pi },\frac{2}{7\pi },\frac{2}{9\pi },\frac{2}{11\pi }\text{,….}$$

The corresponding y-values are

$$1,\mathrm{-1},1,\mathrm{-1},1,\mathrm{-1}\text{,….}$$

At this point we can indeed conclude that $\underset{x\to 0}{\text{lim}}\text{sin}\left(1\text{/}\mathit{\text{x}}\right)$ does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write $\underset{x\to 0}{\text{lim}}\text{sin}\left(1\text{/}\mathit{\text{x}}\right)$ DNE.) The graph of $f\left(x\right)=\text{sin}\left(1\text{/}x\right)$ is shown in Figure 2.17 and it gives a clearer picture of the behavior of $\text{sin}(1\text{/}x)$ as x approaches 0. You can see that $\text{sin}(1\text{/}\mathit{\text{x}})$ oscillates ever more wildly between −1 and 1 as x approaches 0.

Figure 2.17 The graph of $f\left(x\right)=\text{sin}(1\text{/}x)$ oscillates rapidly between −1 and 1 as x approaches 0.

One-Sided Limits

Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function $g\left(x\right)=\left|x-2\right|\text{/}\left(x-2\right)$ introduced at the beginning of the section (see Figure 2.12(b)). As we pick values of x close to 2, $g\left(x\right)$ does not approach a single value, so the limit as x approaches 2 does not exist—that is, $\underset{x\to 2}{\text{lim}}g\left(x\right)$ DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the x-value 2. To provide a more accurate description, we introduce the idea of a one-sided limit. For all values to the left of 2 (or the negative side of 2), $g\left(x\right)=\mathrm{-1}.$ Thus, as x approaches 2 from the left, $g\left(x\right)$ approaches −1. Mathematically, we say that the limit as x approaches 2 from the left is −1. Symbolically, we express this idea as

Similarly, as x approaches 2 from the right (or from the positive side), $g\left(x\right)$ approaches 1. Symbolically, we express this idea as

We can now present an informal definition of one-sided limits.

Example 2.8

Evaluating One-Sided Limits

For the function $f\left(x\right)=\{\begin{array}{ll}x+1& \text{if}x<2\hfill \\ x^2-4& \text{if}x\ge 2\hfill \end{array},$ evaluate each of the following limits.

1. $\underset{x\to 2^{-}}{\text{lim}}f\left(x\right)$
2. $\underset{x\to 2^{+}}{\text{lim}}f\left(x\right)$

Solution

We can use tables of functional values again Table 2.6. Observe that for values of x less than 2, we use $f\left(x\right)=x+1$ and for values of x greater than 2, we use $f\left(x\right)=x^2-4.$

x	$f(x)=x+1$		x	$f(x)=x^2\mathrm{-4}$
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

Table 2.6 Table of Functional Values for $f\left(x\right)=\{\begin{array}{l}x+1\text{if}x<2\\ x^2-4\text{if}x\ge 2\end{array}$

Based on this table, we can conclude that a. $\underset{x\to 2^{-}}{\text{lim}}f\left(x\right)=3$ and b. $\underset{x\to 2^{+}}{\text{lim}}f\left(x\right)=0.$ Therefore, the (two-sided) limit of $f\left(x\right)$ does not exist at $x=2.$ Figure 2.18 shows a graph of $f\left(x\right)$ and reinforces our conclusion about these limits.

Figure 2.18 The graph of $f\left(x\right)=\{\begin{array}{c}x+1\text{if}x<2\\ x^2-4\text{if}x\ge 2\end{array}$ has a break at $x=2.$

Let us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist. These conclusions are summarized in Relating One-Sided and Two-Sided Limits.

Infinite Limits

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.

We now turn our attention to $h\left(x\right)=1\text{/}{(x-2)}^2,$ the third and final function introduced at the beginning of this section (see Figure 2.12(c)). From its graph we see that as the values of x approach 2, the values of $h\left(x\right)=1\text{/}{(x-2)}^2$ become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of $h\left(x\right)$ as x approaches 2 is positive infinity. Symbolically, we express this idea as

More generally, we define infinite limits as follows:

It is important to understand that when we write statements such as $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{+}\infty$ or $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{−}\infty$ we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists. For the limit of a function $f\left(x\right)$ to exist at a, it must approach a real number L as x approaches a. That said, if, for example, $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{+}\infty ,$ we always write $\underset{x\to a}{\text{lim}}f\left(x\right)=\text{+}\infty$ rather than $\underset{x\to a}{\text{lim}}f\left(x\right)$ DNE.

Example 2.9

Recognizing an Infinite Limit

Evaluate each of the following limits, if possible. Use a table of functional values and graph $f\left(x\right)=1\text{/}x$ to confirm your conclusion.

1. $\underset{x\to 0^{-}}{\text{lim}}\frac{1}{x}$
2. $\underset{x\to 0^{+}}{\text{lim}}\frac{1}{x}$
3. $\underset{x\to 0}{\text{lim}}\frac{1}{x}$

Solution

Begin by constructing a table of functional values.

x	$\frac{1}{x}$		x	$\frac{1}{x}$
−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

Table 2.7 Table of Functional Values for $f\left(x\right)=\frac{1}{x}$

1. The values of $1\text{/}x$ decrease without bound as x approaches 0 from the left. We conclude that
   
   $$\underset{x\to 0^{-}}{\text{lim}}\frac{1}{x}=\text{−}\infty .$$
2. The values of $1\text{/}x$ increase without bound as x approaches 0 from the right. We conclude that
   
   $$\underset{x\to 0^{+}}{\text{lim}}\frac{1}{x}=\text{+}\infty .$$
3. Since $\underset{x\to 0^{-}}{\text{lim}}\frac{1}{x}=\text{−}\infty$ and $\underset{x\to 0^{+}}{\text{lim}}\frac{1}{x}=\text{+}\infty$ have different values, we conclude that
   
   $$\underset{x\to 0}{\text{lim}}\frac{1}{x}\text{DNE.}$$

The graph of $f\left(x\right)=1\text{/}x$ in Figure 2.19 confirms these conclusions.

Figure 2.19 The graph of $f\left(x\right)=1\text{/}x$ confirms that the limit as x approaches 0 does not exist.

It is useful to point out that functions of the form $f\left(x\right)=1\text{/}{\left(x-a\right)}^n,$ where n is a positive integer, have infinite limits as x approaches a from either the left or right (Figure 2.20). These limits are summarized in Infinite Limits from Positive Integers.

Figure 2.20 The function $f\left(x\right)=1\text{/}{\left(x-a\right)}^n$ has infinite limits at a.

We should also point out that in the graphs of $f\left(x\right)=1\text{/}{(x-a)}^n,$ points on the graph having x-coordinates very near to a are very close to the vertical line $x=a.$ That is, as x approaches a, the points on the graph of $f\left(x\right)$ are closer to the line $x=a.$ The line $x=a$ is called a vertical asymptote of the graph. We formally define a vertical asymptote as follows:

In the next example we put our knowledge of various types of limits to use to analyze the behavior of a function at several different points.