Finding Limits: Numerical and Graphical Approaches

Learning Outcomes

Find a limit using a graph.
Find a limit using a table.

Understanding Limit Notation

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

[latex]1,\frac{1}{2},\frac{1}{4},\frac{1}{8}..[/latex].

gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function [latex]f\left(x\right)=L[/latex], then as the input [latex]x[/latex] gets closer and closer to [latex]a[/latex], the output y-coordinate gets closer and closer to [latex]L[/latex]. We say that the output “approaches” [latex]L[/latex].

Figure 1 provides a visual representation of the mathematical concept of limit. As the input value [latex]x[/latex] approaches [latex]a[/latex], the output value [latex]f\left(x\right)[/latex] approaches [latex]L[/latex].

Graph representing how a function with a hole at (a, L) approaches a limit.

Figure 1. The output (y-coordinate) approaches [latex]L[/latex] as the input (x-coordinate) approaches [latex]a[/latex].

We write the equation of a limit as

[latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L[/latex].

This notation indicates that as [latex]x[/latex] approaches [latex]a[/latex] both from the left of [latex]x=a[/latex] and the right of [latex]x=a[/latex], the output value approaches [latex]L[/latex].

Consider the function

[latex]f\left(x\right)=\frac{{x}^{2}-6x - 7}{x - 7}[/latex].

We can factor the function as shown.

[latex]\begin{align}&f\left(x\right)=\frac{\cancel{\left(x - 7\right)}\left(x+1\right)}{\cancel{x - 7}}&& \text{Cancel like factors in numerator and denominator.} \\ &f\left(x\right)=x+1,x\ne 7&& \text{Simplify}. \end{align}[/latex]

Notice that [latex]x[/latex] cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition, [latex]x\ne 7[/latex], for the simplified function. We can represent the function graphically as shown in Figure 2.

Graph of an increasing function, f(x) = (x^2-6x-7)/(x-7), with a hole at (7, 8).

Figure 2. Because 7 is not allowed as an input, there is no point at [latex]x=7[/latex].

What happens at [latex]x=7[/latex] is completely different from what happens at points close to [latex]x=7[/latex] on either side. The notation

[latex]\underset{x\to 7}{\mathrm{lim}}f\left(x\right)=8[/latex]

indicates that as the input [latex]x[/latex] approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7.

What happens at [latex]x=7?[/latex] When [latex]x=7[/latex], there is no corresponding output. We write this as

[latex]f\left(7\right)\text{ does not exist}\text{.}[/latex]

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

[latex]f\left(x\right)=x+1,\text{ }x\ne 7[/latex].

Notice that the limit of a function can exist even when [latex]f\left(x\right)[/latex] is not defined at [latex]x=a[/latex]. Much of our subsequent work will be determining limits of functions as [latex]x[/latex] nears [latex]a[/latex], even though the output at [latex]x=a[/latex] does not exist.

A General Note: The Limit of a Function

A quantity [latex]L[/latex] is the limit of a function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] if, as the input values of [latex]x[/latex] approach [latex]a[/latex] (but do not equal [latex]a[/latex]), the corresponding output values of [latex]f\left(x\right)[/latex] get closer to [latex]L[/latex]. Note that the value of the limit is not affected by the output value of [latex]f\left(x\right)[/latex] at [latex]a[/latex]. Both [latex]a[/latex] and [latex]L[/latex] must be real numbers. We write it as

[latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L[/latex]

Example 1: Understanding the Limit of a Function

For the following limit, define [latex]a,f\left(x\right)[/latex], and [latex]L[/latex].

[latex]\underset{x\to 2}{\mathrm{lim}}\left(3x+5\right)=11[/latex]

Show Solution

Try It

For the following limit, define [latex]a,f\left(x\right)[/latex], and [latex]L[/latex].

[latex]\underset{x\to 5}{\mathrm{lim}}\left(2{x}^{2}-4\right)=46[/latex]

Show Solution

Understanding Left-Hand Limits and Right-Hand Limits

We can approach the input of a function from either side of a value—from the left or the right. Figure 3 shows the values of

[latex]f\left(x\right)=x+1,x\ne 7[/latex]

as described earlier and depicted in Figure 2.

Table showing that f(x) approaches 8 from either side as x approaches 7 from either side.

Figure 3

Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in Figure 3 are [latex]6.9[/latex], [latex]6.99[/latex], and [latex]6.999[/latex]. The corresponding outputs are [latex]7.9,7.99[/latex], and [latex]7.999[/latex]. These values are getting closer to 8. The limit of values of [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function [latex]f\left(x\right)=x+1,x\ne 7[/latex] as [latex]x[/latex] approaches 7.

Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in Figure 3 are [latex]7.1[/latex], [latex]7.01[/latex], and [latex]7.001[/latex]. The corresponding outputs are [latex]8.1[/latex], [latex]8.01[/latex], and [latex]8.001[/latex]. These values are getting closer to 8. The limit of values of [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function [latex]f\left(x\right)=x+1,x\ne 7[/latex] as [latex]x[/latex] approaches 7.

Figure 3 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input [latex]x[/latex] within the interval [latex]6.9[latex]\underset{x\to {7}^{-}}{\mathrm{lim}}f\left(x\right)=8[/latex].

To indicate the right-hand limit, we write

[latex]\underset{x\to {7}^{+}}{\mathrm{lim}}f\left(x\right)=8[/latex].

Graph of the previous function explaining the function's limit at (7, 8)

Figure 4. The left- and right-hand limits are the same for this function.

A General Note: Left- and Right-Hand Limits

The left-hand limit of a function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the left is equal to [latex]L[/latex], denoted by

[latex]\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)=L[/latex].

The values of [latex]f\left(x\right)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] such that [latex]xright-hand limit of a function [latex]f\left(x\right)[/latex], as [latex]x[/latex] approaches [latex]a[/latex] from the right, is equal to [latex]L[/latex], denoted by

[latex]\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)=L[/latex].

The values of [latex]f\left(x\right)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] but greater than [latex]a[/latex]. Both [latex]a[/latex] and [latex]L[/latex] are real numbers.

Understanding Two-Sided Limits

In the previous example, the left-hand limit and right-hand limit as [latex]x[/latex] approaches [latex]a[/latex] are equal. If the left- and right-hand limits are equal, we say that the function [latex]f\left(x\right)[/latex] has a two-sided limit as [latex]x[/latex] approaches [latex]a[/latex]. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

A General Note: The Two-Sided Limit of Function as x Approaches a

The limit of a function [latex]f\left(x\right)[/latex], as [latex]x[/latex] approaches [latex]a[/latex], is equal to [latex]L[/latex], that is,

[latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L[/latex]

if and only if

[latex]\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)=\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)[/latex].

In other words, the left-hand limit of a function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] is equal to the right-hand limit of the same function as [latex]x[/latex] approaches [latex]a[/latex]. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.

Finding a Limit Using a Graph

To visually determine if a limit exists as [latex]x[/latex] approaches [latex]a[/latex], we observe the graph of the function when [latex]x[/latex] is very near to [latex]x=a[/latex]. In Figure 5 we observe the behavior of the graph on both sides of [latex]a[/latex].

Graph of a function that explains the behavior of a limit at (a, L) where the function is increasing when x is less than a and decreasing when x is greater than a.

Figure 5

To determine if a left-hand limit exists, we observe the branch of the graph to the left of [latex]x=a[/latex], but near [latex]x=a[/latex]. This is where [latex]xa[/latex]. We see that the outputs are getting close to some real number [latex]L[/latex], so there is a right-hand limit.

If the left-hand limit and the right-hand limit are the same, as they are in Figure 5, then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.

Finally, we can look for an output value for the function [latex]f\left(x\right)[/latex] when the input value [latex]x[/latex] is equal to [latex]a[/latex]. The coordinate pair of the point would be [latex]\left(a,f\left(a\right)\right)[/latex]. If such a point exists, then [latex]f\left(a\right)[/latex] has a value. If the point does not exist, as in Figure 5, then we say that [latex]f\left(a\right)[/latex] does not exist.

How To: Given a function [latex]f\left(x\right)[/latex], use a graph to find the limits and a function value as [latex]x[/latex] approaches [latex]a[/latex].

Examine the graph to determine whether a left-hand limit exists.
Examine the graph to determine whether a right-hand limit exists.
If the two one-sided limits exist and are equal, then there is a two-sided limit—what we normally call a “limit.”
If there is a point at [latex]x=a[/latex], then [latex]f\left(a\right)[/latex] is the corresponding function value.

Example 2: Finding a Limit Using a Graph

Determine the following limits and function value for the function [latex]f[/latex] shown in Figure 6.
1. [latex]\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)[/latex]
2. [latex]\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)[/latex]
3. [latex]\underset{x\to 2}{\mathrm{lim}}f\left(x\right)[/latex]
4. [latex]f\left(2\right)[/latex]
Figure 6
Determine the following limits and function value for the function [latex]f[/latex] shown in Figure 7.
1. [latex]\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)[/latex]
2. [latex]\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)[/latex]
3. [latex]\underset{x\to 2}{\mathrm{lim}}f\left(x\right)[/latex]
4. [latex]f\left(2\right)[/latex]
Figure 7
Show Solution
1. Looking at Figure 6:
  1. [latex]\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=8[/latex]; when [latex]x<2[/latex], but infinitesimally close to 2, the output values get close to [latex]y=8[/latex].
  2. [latex]\underset{x\to 2{}^{+}}{\mathrm{lim}}f\left(x\right)=3[/latex]; when [latex]x>2[/latex], but infinitesimally close to 2, the output values approach [latex]y=3[/latex].
  3. [latex]\underset{x\to 2}{\mathrm{lim}}f\left(x\right)[/latex] does not exist because [latex]\underset{x\to 2{}^{-}}{\mathrm{lim}}f\left(x\right)\ne \underset{x\to 2{}^{+}}{\mathrm{lim}}f\left(x\right)[/latex]; the left and right-hand limits are not equal.
  4. [latex]f\left(2\right)=3[/latex] because the graph of the function [latex]f[/latex] passes through the point [latex]\left(2,f\left(2\right)\right)[/latex] or [latex]\left(2,3\right)[/latex].
2. Looking at Figure 7:
  1. [latex]\underset{x\to 2{}^{-}}{\mathrm{lim}}f\left(x\right)=8[/latex]; when [latex]x<2[/latex] but infinitesimally close to 2, the output values approach [latex]y=8[/latex].
  2. [latex]\underset{x\to 2{}^{+}}{\mathrm{lim}}f\left(x\right)=8[/latex]; when [latex]x>2[/latex] but infinitesimally close to 2, the output values approach [latex]y=8[/latex].
  3. [latex]\underset{x\to 2}{\mathrm{lim}}f\left(x\right)=8[/latex] because [latex]\underset{x\to 2{}^{-}}{\mathrm{lim}}f\left(x\right)=\underset{x\to 2{}^{+}}{\mathrm{lim}}f\left(x\right)=8[/latex]; the left and right-hand limits are equal.
  4. [latex]f\left(2\right)=4[/latex] because the graph of the function [latex]f[/latex] passes through the point [latex]\left(2,f\left(2\right)\right)[/latex] or [latex]\left(2,4\right)[/latex].

Try It

Using the graph of the function [latex]y=f\left(x\right)[/latex] shown in Figure 8, estimate the following limits.

Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)

Figure 8

Show Solution

Try It

Finding a Limit Using a Table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of [latex]x[/latex] approach [latex]a[/latex] from both sides. Then we determine if the output values get closer and closer to some real value, the limit [latex]L[/latex].

Let’s consider an example using the following function:

[latex]\underset{x\to 5}{\mathrm{lim}}\left(\dfrac{{x}^{3}-125}{x - 5}\right)[/latex]

To create the table, we evaluate the function at values close to [latex]x=5[/latex]. We use some input values less than 5 and some values greater than 5 as in Figure 9. The table values show that when [latex]x>5[/latex] but nearing 5, the corresponding output gets close to 75. When [latex]x>5[/latex] but nearing 5, the corresponding output also gets close to 75.

Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.

Figure 9

Because

[latex]\underset{x\to {5}^{-}}{\mathrm{lim}}f\left(x\right)=75=\underset{x\to {5}^{+}}{\mathrm{lim}}f\left(x\right)[/latex],

then

[latex]\underset{x\to 5}{\mathrm{lim}}f\left(x\right)=75[/latex].

Remember that [latex]f\left(5\right)[/latex] does not exist.

How To: Given a function [latex]f[/latex], use a table to find the limit as [latex]x[/latex] approaches [latex]a[/latex] and the value of [latex]f\left(a\right)[/latex], if it exists.

Choose several input values that approach [latex]a[/latex] from both the left and right. Record them in a table.
Evaluate the function at each input value. Record them in the table.
Determine if the table values indicate a left-hand limit and a right-hand limit.
If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.
Replace [latex]x[/latex] with [latex]a[/latex] to find the value of [latex]f\left(a\right)[/latex].

Example 3: Finding a Limit Using a Table

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

[latex]\underset{x\to 0}{\mathrm{lim}}\left(\dfrac{5\sin \left(x\right)}{3x}\right)[/latex]

Show Solution

Q & A

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

Yes. We previously used a table to find a limit of 75 for the function [latex]f\left(x\right)=\frac{{x}^{3}-125}{x - 5}[/latex] as [latex]x[/latex] approaches 5. To check, we graph the function on a viewing window as shown in Figure 11. A graphical check shows both branches of the graph of the function get close to the output 75 as [latex]x[/latex] nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By appraoching [latex]x=5[/latex] we may numerically observe the corresponding outputs getting close to [latex]75[/latex].

Graph of an increasing function with a discontinuity at (5, 75)

Figure 11

Try It

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

[latex]\underset{x\to 0}{\mathrm{lim}}\left(\dfrac{20\sin \left(x\right)}{4x}\right)[/latex]

Show Solution

Try It

Q & A

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

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

Example 4: Using a Graphing Utility to Determine a Limit

With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as [latex]x[/latex] approaches 0. If the function has a limit as [latex]x[/latex] approaches 0, state it. If not, discuss why there is no limit.

[latex]f\left(x\right)=3\sin \left(\frac{\pi }{x}\right)[/latex]

Show Solution

Try It

Numerically estimate the following limit: [latex]\underset{x\to 0}{\mathrm{lim}}\left(\sin \left(\dfrac{2}{x}\right)\right)[/latex].

Show Solution

Key Concepts

A function has a limit if the output values approach some value [latex]L[/latex] as the input values approach some quantity [latex]a[/latex].
A shorthand notation is used to describe the limit of a function according to the form [latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L[/latex], which indicates that as [latex]x[/latex] approaches [latex]a[/latex], both from the left of [latex]x=a[/latex] and the right of [latex]x=a[/latex], the output value gets close to [latex]L[/latex].
A function has a left-hand limit if [latex]f\left(x\right)[/latex] approaches [latex]L[/latex] as [latex]x[/latex] approaches [latex]a[/latex] where [latex]xa[/latex].
A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
A graph provides a visual method of determining the limit of a function.
If the function has a limit as [latex]x[/latex] approaches [latex]a[/latex], the branches of the graph will approach the same [latex]y\text{-}[/latex] coordinate near [latex]x=a[/latex] from the left and the right.
A table can be used to determine if a function has a limit. The table should show input values that approach [latex]a[/latex] from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit.
A graphing utility can also be used to find a limit.

Glossary

left-hand limit: the limit of values of [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches from [latex]a[/latex] the left, denoted [latex]\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)=L[/latex]. The values of [latex]f\left(x\right)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] such that [latex]x

limit: when it exists, the value, [latex]L[/latex], that the output of a function [latex]f\left(x\right)[/latex] approaches as the input [latex]x[/latex] gets closer and closer to [latex]a[/latex] but does not equal [latex]a[/latex]. The value of the output, [latex]f\left(x\right)[/latex], can get as close to [latex]L[/latex] as we choose to make it by using input values of [latex]x[/latex] sufficiently near to [latex]x=a[/latex], but not necessarily at [latex]x=a[/latex]. Both [latex]a[/latex] and [latex]L[/latex] are real numbers, and [latex]L[/latex] is denoted [latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L[/latex].

right-hand limit: the limit of values of [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the right, denoted [latex]\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)=L[/latex]. The values of [latex]f\left(x\right)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] where [latex]x>a[/latex], and [latex]x\ne a[/latex]. Both [latex]a[/latex] and [latex]L[/latex] are real numbers.

two-sided limit: the limit of a function [latex]f\left(x\right)[/latex], as [latex]x[/latex] approaches [latex]a[/latex], is equal to [latex]L[/latex], that is, [latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=L[/latex] if and only if [latex]\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)=\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)[/latex].

Module 12: Introduction to Calculus