In this section, you will:
- Understand limit notation.
- Find a limit using a graph.
- Find a limit using a table.
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.1 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.
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
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 y-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
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 factor the function as shown.
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.
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.
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 it as
Understanding the Limit of a Function
For the following limit, define and
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
For the following limit, define and
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
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 Figure 3 are and The corresponding outputs are and 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.
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 and The corresponding outputs are and 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.
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 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 4 provides a visual representation of the left- 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
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.
Understanding Two-Sided Limits
In the previous example, the left-hand limit and right-hand limit as approaches are equal. If the left- 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 limit of a function as approaches is equal to that is,
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.
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 5 we observe the behavior of the graph on both sides of
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 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 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 5, then we say that does not exist.
Given a function use a graph to find the limits and a function value as approaches
- 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 then is the corresponding function value.
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 approach from both sides. Then we determine if the output values get closer and closer to some real value, the limit
Let’s consider an example using the following function:
To create the table, we evaluate the function at values close to We use some input values less than 5 and some values greater than 5 as in Figure 9. The table values show that when but nearing 5, the corresponding output gets close to 75. When but nearing 5, the corresponding output also gets close to 75.
Remember that does not exist.
Given a function use a table to find the limit as approaches and the value of if it exists.
- Choose several input values that approach 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 with to find the value of
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.
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 as 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 nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By appraoching we may numerically observe the corresponding outputs getting close to
Numerically estimate the limit of the following function by making a table:
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.
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 approaches 0. If the function has a limit as approaches 0, state it. If not, discuss why there is no limit.
Numerically estimate the following limit:
12.1 Section Exercises
Explain the difference between a value at and the limit as approaches
Explain why we say a function does not have a limit as approaches if, as approaches the left-hand limit is not equal to the right-hand limit.
For the following exercises, estimate the functional values and the limits from the graph of the function provided in Figure 14.
For the following exercises, draw the graph of a function from the functional values and limits provided.
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For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as approaches 0.
Based on the pattern you observed in the exercises above, make a conjecture as to the limit of
For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as approaches If the function has a limit as approaches state it. If not, discuss why there is no limit.
For the following exercises, use numerical evidence to determine whether the limit exists at If not, describe the behavior of the graph of the function near Round answers to two decimal places.
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as approaches the given value.
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as approaches If the function has a limit as approaches state it. If not, discuss why there is no limit.
Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: and as approaches 0. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions and as approaches 0. If the functions have a limit as approaches 0, state it. If not, discuss why there is no limit.
According to the Theory of Relativity, the mass of a particle depends on its velocity . That is
where is the mass when the particle is at rest and is the speed of light. Find the limit of the mass, as approaches
Allow the speed of light, to be equal to 1.0. If the mass, is 1, what occurs to as Using the values listed in Table 1, make a conjecture as to what the mass is as approaches 1.00.
- 1https://en.wikipedia.org/wiki/Human_height and http://en.wikipedia.org/wiki/List_of_tallest_people