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Finding Limits: Numerical and Graphical Approaches

Jay Abramson

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

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

1. Understanding Limit Notation

We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the number of terms increases. For example, the terms of the sequence 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 , then as the input gets closer and closer to , the output -coordinate gets closer and closer to . We say that the output “approaches” .

Figure 1 provides a visual representation of the mathematical concept of limit. As the input value approaches , the output value approaches .

Figure 1: The output (-coordinate) approaches as the input (-coordinate) approaches .

We write the equation of a limit as This notation indicates that as approaches both from the left of and the right of , the output value approaches .

Consider the function We can simplify this function, by factoring and canceling as shown: Thus, we have , where . Notice that 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, , for the simplified function. We can represent the function graphically as shown in Figure 2.

Figure 2: A graph representing the function . Because 7 is not allowed as an input, there is no point at .

What happens at is completely different from what happens at points close to on either side. The notation indicates that as the input 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 ? When , there is no corresponding output. We write this as This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as Notice that the limit of a function can exist even when is not defined at . Much of our subsequent work will be determining limits of functions as nears , even though the output at does not exist.

The Limit of a Function

A quantity is the limit of a function as approaches if, as the input values of approach (but do not equal ), the corresponding output values of get closer to . Note that the value of the limit is not affected by the output value of at . Both and must be real numbers. We write the limit as approaches a of in the following way:

Example 1: Understanding the Limit of a Function

For the limit define , the number is approaching, , and the limit .

Solution

First, we recognize the notation of a limit. If the limit exists, as approaches , we write We are given This means that , , and .

Analysis

Recall that is a line with no breaks. As the input values approach 2, the output values will get close to 11. This may be phrased with the equation , which means that as nears 2 (but is not exactly 2), the output of the function gets as close as we want to , or 11, which is the limit , as we take values of sufficiently near 2 but not at .

1.1. 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. Table 1 shows the values of as described earlier and depicted in Figure 2.

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 Table 1 are 6.9, 6.99, and 6.999. The corresponding outputs are 7.9, 7.99, and 7.999. These values are getting closer to 8. The limit of values of as approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function as approaches 7.

Table 1
Values of approach 7 from the left Values of approach 7 from the right
6.9 6.99 6.999 7 7.001 7.01 7.1
7.9 7.99 7.999 Undefined 8.001 8.01 8.1

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 Table 1 are 7.1, 7.01, and 7.001. The corresponding outputs are 8.1, 8.01, and 8.001. These values are getting closer to 8. The limit of values of as approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function as approaches 7.

Table 1 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 within the interval to produce an output value of within the interval .

We also see that we can get output values of successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.

Figure 3 provides a visual representation of the left-hand and right-hand limits of the function. From the graph of , we observe the output can get infinitesimally close to as approaches 7 from the left and as approaches 7 from the right.

To indicate the left-hand limit, we write To indicate the right-hand limit, we write

Figure 3: The left-hand and right-hand limits are the same for this function.

Left-Hand and Right-Hand Limits

The left-hand limit of a function as approaches from the left is equal to , denoted by The values of can get as close to the limit as we like by taking values of sufficiently close to such that and .

The right-hand limit of a function , as approaches from the right, is equal to , denoted by The values of can get as close to the limit as we like by taking values of sufficiently close to but greater than . Both and are real numbers.

1.2. Understanding Two-Sided Limits

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

The Two-Sided Limit of Function as Approaches

The limit of a function , as approaches , is equal to if and only if In other words, the left-hand limit of a function as approaches is equal to the right-hand limit of the same function as approaches . If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.

2. Finding a Limit Using a Graph

To visually determine if a limit exists as approaches , we observe the graph of the function when is very near to . In Figure 4 we observe the behavior of the graph on both sides of .

Figure 4: Determining the limit of a function using a graph.

To determine if a left-hand limit exists, we observe the branch of the graph to the left of , but near . This is where . We see that the outputs are getting close to some real number so there is a left-hand limit.

To determine if a right-hand limit exists, observe the branch of the graph to the right of , but near . This is where . We see that the outputs are getting close to some real number , 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 4, 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 when the input value is equal to . The coordinate pair of the point would be . If such a point exists, then has a value. If the point does not exist, as in Figure 4, then we say that does not exist.

How To …

Given a function , use a graph to find the limits and a function value as approaches .

  1. Examine the graph to determine whether a left-hand limit exists.
  2. Examine the graph to determine whether a right-hand limit exists.
  3. If the two one-sided limits exist and are equal, then there is a two-sided limit—what we normally call a “limit.”
  4. If there is a point at , then is the corresponding function value.

Example 2: Finding a Limit Using a Graph

  1. Determine the following limits and function value for the function shown in Figure 5.
  2. Determine the following limits and function value for the function shown in Figure 6.

Figure 5
Figure 6

Solution

  1. Looking at Figure 5 we see that
    1. ; when , but infinitesimally close to 2, the output values get close to .
    2. ; when , but infinitesimally close to 2, the output values approach .
    3. does not exist because ; the left-hand and right-hand limits are not equal.
    4. because the graph of the function passes through the point or