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Section 12.4 Limits at Infinity and Limits of Sequences 883
12.4 Limits at Infinity and Limits of Sequences
Whatyou should learn Limits at Infinity and Horizontal Asymptotes
•Evaluate limits of functions
at infinity. As pointed out at the beginning of this chapter, there are two basic problems in
•Find limits of sequences. calculus: finding tangent lines and finding the area of a region. In Section 12.3,
Whyyou should learn it you saw how limits can be used to solve the tangent line problem. In this section
and the next, you will see how a different type of limit, a limit at infinity, can be
Finding limits at infinity is used to solve the area problem. To get an idea of what is meant by a limit at
useful in many types of real-life infinity, consider the function given by
applications.For instance,in
Exercise 52 on page 891,you f x x 1.
are asked to find a limit at infinity 2x
to determine the number of The graph of f is shown in Figure 12.30. From earlier work, you know that y 1
military reserve personnel in the 2
future. is a horizontal asymptote of the graph of this function. Using limit notation, this
can be written as follows.
lim f x 1 Horizontal asymptote to the left
x→ 2
lim f x 1 Horizontal asymptote to the right
x→ 2
1
These limits mean that the value of fx gets arbitrarily close to as x decreases
2
or increases without bound.
y
3 x + 1
©Karen Kasmauski/Corbis f(x) = 2x
1 2
y =
2 1
x
3 2 1 123
2
3
FIGURE 12.30
Definition of Limits at Infinity
If f is a function and L and L are real numbers, the statements
1 2
lim f x L Limit as x approaches
x→ 1
and
lim f x L2 Limit as x approaches
x→
denote the limits at infinity. The first statement is read “the limit of fx as
x L , f x x
approaches is ” and the second is read “the limit of as
1
approaches is L .”
2
332522_1204.qxd 12/13/05 1:06 PM Page 884
884 Chapter 12 Limits and an Introduction to Calculus
Exploration To help evaluate limits at infinity, you can use the following definition.
Use a graphing utility to graph Limits at Infinity
the two functions given by If r is a positive real number, then
y 1 and y 1 1
1 2 3
x x lim r 0. Limit toward the right
x→ x
in the same viewing window. r
Why doesn’t y appear to the Furthermore, if x is defined when x < 0, then
1
left of the y-axis? How does lim 1 0. Limit toward the left
this relate to the statement at x→xr
the right about the infinite limit
lim 1 ? Limits at infinity share many of the properties of limits listed in Section
x→ xr 12.1. Some of these properties are demonstrated in the next example.
Example 1 Evaluating a Limit at Infinity
Find the limit.
lim 4 3
x→ 2
x
Algebraic Solution Graphical Solution
2
Use the properties of limits listed in Section 12.1. Use a graphing utility to graph y 4 3x . Then use the
3 3 trace feature to determine that as x gets larger and larger, y gets
lim 4 2 lim 4 lim 2 closer and closer to 4, as shown in Figure 12.31. Note that the
x→ x x→ x→x line y 4 is a horizontal asymptote to the right.
lim 4 3lim 1
x→ x→x2 5 y = 4
4 30
4 y = 4 3
x2
So, the limit of fx 4 3 as x approaches is 4. 20 120
x2
1
Now try Exercise 5. FIGURE 12.31
In Figure 12.31, it appears that the line y 4 is also a horizontal asymptote
to the left. You can verify this by showing that
lim 4 3 4.
x→ x2
The graph of a rational function need not have a horizontal asymptote. If it does,
however, its left and right horizontal asymptotes must be the same.
When evaluating limits at infinity for more complicated rational functions,
divide the numerator and denominator by the highest-powered term in the denom-
inator. This enables you to evaluate each limit using the limits at infinity at the
top of the page.
332522_1204.qxd 12/13/05 1:06 PM Page 885
Section 12.4 Limits at Infinity and Limits of Sequences 885
Example 2
Exploration Comparing Limits at Infinity
Find the limit as x approaches for each function.
Use a graphing utility to
complete the table below to 2x3 2x2 3 2x3 3
verify that a. fx 3x2 1 b. fx 3x2 1 c. fx 3x2 1
lim 1 0. Solution
x→ x
In each case, begin by dividing both the numerator and denominator by 2 the
x ,
0 1 2 highest-powered term in the denominator.
x 10 10 10
1 2 3
x 2x3 x x2
a. lim 2 lim
x→ 3x 1 x→ 1
3 4 5 3 2
x 10 10 10 x
1 0 0
x 3 0
0
Make a conjecture about 3
1 2x2 3 2x2
lim . b. lim lim
x→0 x x→ 2 x→
3x 1 3 1
x2
2 0
3 0
2
3
2x 3
3 2
c. lim 2x 3 lim x
x→ 2 x→
3x 1 3 1
x2
Have students use these observations In this case, you can conclude that the limit does not exist because the
from Example 2 to predict the following numerator decreases without bound as the denominator approaches 3.
limits.
5xx 3 Now try Exercise 13.
a. lim 2x
x→
4x3 5x In Example 2, observe that when the degree of the numerator is less than the
b. lim 4 2
x→ 8x 3x 2
degree of the denominator, as in part (a), the limit is 0. When the degrees of the
2
6x 1 numerator and denominator are equal, as in part (b), the limit is the ratio of the
c. lim 2
x→ 3x x 2 coefficients of the highest-powered terms. When the degree of the numerator is
Then ask several students to verify the greater than the degree of the denominator, as in part (c), the limit does not exist.
predictions algebraically,several other This result seems reasonable when you realize that for large values of x, the
students to verify the predictions highest-powered term of a polynomial is the most “influential” term. That is, a
numerically,and several more students polynomial tends to behave as its highest-powered term behaves as x approaches
to verify the predictions graphically.
Lead a discussion comparing the results. positive or negative infinity.
332522_1204.qxd 12/13/05 1:06 PM Page 886
886 Chapter 12 Limits and an Introduction to Calculus
Limits at Infinity for Rational Functions
Consider the rational function fx NxDx, where
n . . . m . . .
Nx a x a and Dx b x b.
n 0 m 0
The limit of fx as x approaches positive or negative infinity is as follows.
0, n < m
lim fx an
x→± b , n m
m
If n > m, the limit does not exist.
Example 3 Finding the Average Cost
Consider asking your students to You are manufacturing greeting cards that cost $0.50 per card to produce. Your
identify the practical interpretation initial investment is $5000, which implies that the total cost C of producing x
of the limit in part (d) of Example 3. cards is given by C 0.50x 5000. The average cost C per card is given by
CC0.50x5000.
x x
Find the average cost per card when (a) x 1000, (b) x 10,000, and
(c) x 100,000. (d) What is the limit of C as x approaches infinity?
Solution
a. When x 1000, the average cost per card is
C 0.501000 5000 x 1000
1000
$5.50.
b. When x 10,000, the average cost per card is
Average Cost 0.5010,000 5000
C C x 10,000
10,000
6 $1.00.
5 c. When x 100,000, the average cost per card is
4 0.50100,000 5000
3 C 0.50x + 5000 C x 100,000
C = = 100,000
(in dollars) x x
erage cost per card 2 $0.55.
Av 1 d. As x approaches infinity, the limit of C is
x 0.50x 5000
y = 0.5 20,000 60,000 100,000 lim $0.50. x →
x→
Number of cards x
As x → , the average cost per card
The graph of C is shown in Figure 12.32.
approaches $0.50.
FIGURE 12.32 Now try Exercise 49.
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