5 Steps to a 5 AP Calculus BC 2019

Limits and Continuity 63

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 5.2-8.)

[–5,5] by [–30,30]
Figure 5.2-8
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)=x→lim−∞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 61.

(2) If the degree ofp(x) is smaller than the degree ofq(x), then limx→∞f(x)=xlim→−∞f(x)=0.

The liney=0 (orx-axis) is a horizontal asymptote. See Example 2 on page 62.

(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 62.

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 5.2-9.)

[–20,20] by [–3,3]

Figure 5.2-9

TIP • When entering a rational function into a calculator, use parentheses for both the

numerator and denominator, e.g., (x−2)+(x+3).