MA 3972-MA-Book May 7, 2018 9:52
98 STEP 4. Review the Knowledge You Need to Score High
you notice that limx→ 0 +f(x)=∞, and limx→ 0 −f(x)=−∞. Therefore, the linex=0 (or they-axis)
is a vertical asymptote. (See Figure 6.2-8.)
[−5, 5] by [−30, 30]
Figure 6.2-8
Relationship between the limits of rational functions asx→∞and horizontal asymptotes:
KEY IDEA
Givenf(x)=
p(x)
q(x)
, then:
(1) If the degree ofp(x) is the same as the degree ofq(x), then limx→∞f(x)=xlim→−∞f(x)=
a
b
,
whereais the coefficient of the highest power ofxinp(x) andbis the coefficient of
the highest power ofxinq(x). The liney=
a
b
is a horizontal asymptote. See Example 1
on page 96.
(2) If the degree ofp(x) is smaller than the degree ofq(x), then limx→∞f(x)=x→lim−∞f(x)=0.
The liney=0 (or thex-axis) is a horizontal asymptote. See Example 2 on page 97.
(3) If the degree of p(x) is greater than the degree ofq(x), then limx→∞f(x)=±∞
and limx→−∞f(x)=±∞. Thus, f(x) has no horizontal asymptote. (See Example 3
on page 97.)
Example 4
Using your calculator, find the horizontal asymptotes of the functionf(x)=
2 sinx
x
.
Entery 1 =
2 sinx
x
. The graph shows thatf(x) oscillates back and forth about thex-axis. As
x→±∞, the graph gets closer and closer to thex-axis, which implies thatf(x) approaches
0. Thus, the liney=0 (or thex-axis) is a horizontal asymptote. (See Figure 6.2-9.)
[−20, 20] by [−3, 3]
Figure 6.2-9
TIP • When entering a rational function into a calculator, use parentheses for both the
numerator and denominator, e.g., (x−2)+(x+3).