14
Learning Objectives
- Calculate the limit of a function as increases or decreases without bound.
- Recognize a horizontal asymptote on the graph of a function.
- Estimate the end behavior of a function as increases or decreases without bound.
- Recognize an oblique asymptote on the graph of a function.
To graph a function defined on an unbounded domain, we also need to know the behavior of as In this section, we define limits at infinity and show how these limits affect the graph of a function.
Limits at Infinity
We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. Back in Introduction to Functions and Graphs, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.
Limits at Infinity and Horizontal Asymptotes
Recall that means becomes arbitrarily close to as long as is sufficiently close to We can extend this idea to limits at infinity. For example, consider the function As can be seen graphically in (Figure) and numerically in (Figure), as the values of get larger, the values of approach 2. We say the limit as approaches of is 2 and write Similarly, for as the values get larger, the values of approaches 2. We say the limit as approaches of is 2 and write
10 | 100 | 1,000 | 10,000 | |
2.1 | 2.01 | 2.001 | 2.0001 | |
-10 | -100 | -1000 | -10,000 | |
1.9 | 1.99 | 1.999 | 1.9999 |
More generally, for any function we say the limit as of is if becomes arbitrarily close to as long as is sufficiently large. In that case, we write Similarly, we say the limit as of is if becomes arbitrarily close to as long as and is sufficiently large. In that case, we write We now look at the definition of a function having a limit at infinity.
Definition
(Informal) If the values of become arbitrarily close to as becomes sufficiently large, we say the function has a limit at infinity and write
If the values of becomes arbitrarily close to for as becomes sufficiently large, we say that the function has a limit at negative infinity and write
If the values are getting arbitrarily close to some finite value as or the graph of approaches the line In that case, the line is a horizontal asymptote of ((Figure)). For example, for the function since the line is a horizontal asymptote of
Definition
If or we say the line is a horizontal asymptote of
A function cannot cross a vertical asymptote because the graph must approach infinity (or from at least one direction as approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function shown in (Figure) intersects the horizontal asymptote an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.
The algebraic limit laws and squeeze theorem we introduced in Introduction to Limits also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.
Computing Limits at Infinity
For each of the following functions evaluate and Determine the horizontal asymptote(s) for
Solution
- Using the algebraic limit laws, we have Similarly, Therefore, has a horizontal asymptote of and approaches this horizontal asymptote as as shown in the following graph.
- Since for all we have
for all Also, since
we can apply the squeeze theorem to conclude that
Similarly,
Thus, has a horizontal asymptote of and approaches this horizontal asymptote as as shown in the following graph.
- To evaluate and we first consider the graph of over the interval as shown in the following graph.
Since
it follows that
Similarly, since
it follows that
As a result, and are horizontal asymptotes of as shown in the following graph.
Evaluate and Determine the horizontal asymptotes of if any.
Hint
Solution
Both limits are 3. The line is a horizontal asymptote.
Infinite Limits at Infinity
Sometimes the values of a function become arbitrarily large as (or as In this case, we write (or On the other hand, if the values of are negative but become arbitrarily large in magnitude as (or as we write (or
For example, consider the function As seen in (Figure) and (Figure), as the values become arbitrarily large. Therefore, On the other hand, as the values of are negative but become arbitrarily large in magnitude. Consequently,
10 | 20 | 50 | 100 | 1000 | |
1000 | 8000 | 125,000 | 1,000,000 | 1,000,000,000 | |
-10 | -20 | -50 | -100 | -1000 | |
-1000 | -8000 | -125,000 | -1,000,000 | -1,000,000,000 |
Definition
(Informal) We say a function has an infinite limit at infinity and write
if becomes arbitrarily large for sufficiently large. We say a function has a negative infinite limit at infinity and write
if and becomes arbitrarily large for sufficiently large. Similarly, we can define infinite limits as
Formal Definitions
Earlier, we used the terms arbitrarily close, arbitrarily large, and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.
Definition
(Formal) We say a function has a limit at infinity, if there exists a real number such that for all there exists such that
for all In that case, we write
(see (Figure)).
We say a function has a limit at negative infinity if there exists a real number such that for all there exists such that
for all In that case, we write
Earlier in this section, we used graphical evidence in (Figure) and numerical evidence in (Figure) to conclude that Here we use the formal definition of limit at infinity to prove this result rigorously.
A Finite Limit at Infinity Example
Use the formal definition of limit at infinity to prove that
Solution
Let Let Therefore, for all we have
Use the formal definition of limit at infinity to prove that
Hint
Let
Solution
Let Let Therefore, for all we have
Therefore,
We now turn our attention to a more precise definition for an infinite limit at infinity.
Definition
(Formal) We say a function has an infinite limit at infinity and write
if for all there exists an such that
for all (see (Figure)).
We say a function has a negative infinite limit at infinity and write
if for all there exists an such that
for all
Similarly we can define limits as
Earlier, we used graphical evidence ((Figure)) and numerical evidence ((Figure)) to conclude that Here we use the formal definition of infinite limit at infinity to prove that result.
An Infinite Limit at Infinity
Use the formal definition of infinite limit at infinity to prove that
Solution
Let Let Then, for all we have
Therefore,
Use the formal definition of infinite limit at infinity to prove that
Hint
Let
Solution
Let Let Then, for all we have
End Behavior
The behavior of a function as is called the function’s end behavior. At each of the function’s ends, the function could exhibit one of the following types of behavior:
- The function approaches a horizontal asymptote
- The function or
- The function does not approach a finite limit, nor does it approach or In this case, the function may have some oscillatory behavior.
Let’s consider several classes of functions here and look at the different types of end behaviors for these functions.
End Behavior for Polynomial Functions
Consider the power function where is a positive integer. From (Figure) and (Figure), we see that
and
Using these facts, it is not difficult to evaluate and where is any constant and is a positive integer. If the graph of is a vertical stretch or compression of and therefore
If the graph of is a vertical stretch or compression combined with a reflection about the -axis, and therefore
If in which case
Limits at Infinity for Power Functions
For each function evaluate and
Solution
- Since the coefficient of is -5, the graph of involves a vertical stretch and reflection of the graph of about the -axis. Therefore, and
- Since the coefficient of is 2, the graph of is a vertical stretch of the graph of Therefore, and
- This is an form so limit laws don’t help. However, by factoring we get
As , the power where the second factor gets close to . Hence the limit is As , the power where the second factor gets close to . Hence the limit is
Let Find
Hint
The coefficient -3 is negative.
Solution
End Behavior for Algebraic Functions
The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of
Determining End Behavior for Rational Functions
For each of the following functions, determine the limits as and Then, use this information to describe the end behavior of the function.
Solution
- The highest power of in the denominator is Therefore, dividing the numerator and denominator by and applying the algebraic limit laws, we see that
Since we know that is a horizontal asymptote for this function as shown in the following graph.
- Since the largest power of appearing in the denominator is divide the numerator and denominator by After doing so and applying algebraic limit laws, we obtain
Therefore has a horizontal asymptote of as shown in the following graph.
- Dividing the numerator and denominator by we have
As the denominator approaches 1. As the numerator approaches As the numerator approaches Therefore whereas as shown in the following figure.
Evaluate and use these limits to determine the end behavior of
Hint
Divide the numerator and denominator by
Solution
Before proceeding, consider the graph of shown in (Figure). As and the graph of appears almost linear. Although is certainly not a linear function, we now investigate why the graph of seems to be approaching a linear function. First, using long division of polynomials, we can write
Since as we conclude that
Therefore, the graph of approaches the line as This line is known as an oblique asymptote for ((Figure)).
Now let’s consider the end behavior for functions involving a radical.
Determining End Behavior for a Function Involving a Radical
Find the limits as and for and describe the end behavior of
Solution
Let’s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of To determine the appropriate power of consider the expression in the denominator. Since
for large values of in effect appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by Then, using the fact that for for and for all we calculate the limits as follows:
Therefore, approaches the horizontal asymptote as and the horizontal asymptote as as shown in the following graph.
Evaluate
Hint
Divide the numerator and denominator by
Solution
Determining End Behavior for Transcendental Functions
The six basic trigonometric functions are periodic and do not approach a finite limit as For example, oscillates between ((Figure)). The tangent function has an infinite number of vertical asymptotes as therefore, it does not approach a finite limit nor does it approach as as shown in (Figure).
Recall that for any base the function is an exponential function with domain and range If is increasing over If is decreasing over For the natural exponential function Therefore, is increasing on and the range is The exponential function approaches as and approaches 0 as as shown in (Figure) and (Figure).
-5 | -2 | 0 | 2 | 5 | |
0.00674 | 0.135 | 1 | 7.389 | 148.413 |
Recall that the natural logarithm function is the inverse of the natural exponential function Therefore, the domain of is and the range is The graph of is the reflection of the graph of about the line Therefore, as and as as shown in (Figure) and (Figure).
0.01 | 0.1 | 1 | 10 | 100 | |
-4.605 | -2.303 | 0 | 2.303 | 4.605 |
Determining End Behavior for a Transcendental Function
Find the limits as and for and describe the end behavior of
Solution
To find the limit as divide the numerator and denominator by
As shown in (Figure), as Therefore,
We conclude that and the graph of approaches the horizontal asymptote as To find the limit as use the fact that as to conclude that and therefore the graph of approaches the horizontal asymptote as
Find the limits as and for
Hint
and
Solution
,
Key Concepts
- The limit of is as (or as if the values become arbitrarily close to as becomes sufficiently large.
- The limit of is as if becomes arbitrarily large as becomes sufficiently large. The limit of is as if and becomes arbitrarily large as becomes sufficiently large. We can define the limit of as approaches similarly.
For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.
Solution
Solution
Solution
For the following functions determine whether there is an asymptote at Justify your answer without graphing on a calculator.
6.
7.
Solution
Yes, there is a vertical asymptote
8.
9.
Solution
Yes, there is vertical asymptote
10.
For the following exercises, evaluate the limit.
11.
Solution
0
12.
13.
Solution
14.
15.
Solution
16.
17.
Solution
-2
18.
19.
Solution
-4
20.
21.
Solution
22.
23.
Solution
24.
For the following exercises, find the horizontal and vertical asymptotes.
25.
Solution
Horizontal: none, vertical:
26.
27.
Solution
Horizontal: none, vertical:
28.
29.
Solution
Horizontal: none, vertical: none
30.
31.
Solution
Horizontal: vertical:
32.
33.
Solution
Horizontal: vertical: and
34.
35.
Solution
Horizontal: vertical:
36.
37.
Solution
Horizontal: none, vertical: none
38.
39.
Solution
Horizontal: , vertical:
40.
For the following exercises, construct a function that has the given asymptotes.
41. and
Solution
Answers will vary, for example:
42. and
43. and
Solution
Answers will vary, for example:
44.
For the following exercises, graph the function on a graphing calculator on the window and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.
45. [T]
Solution
46. [T]
47. [T]
Solution
48. [T]
49. [T]
Solution
50. True or false: Every ratio of polynomials has vertical asymptotes.
Glossary
- end behavior
- the behavior of a function as and
- horizontal asymptote
- if or then is a horizontal asymptote of
- infinite limit at infinity
- a function that becomes arbitrarily large as becomes large
- limit at infinity
- the limiting value, if it exists, of a function as or
- oblique asymptote
- the line if approaches it as or