5 Steps to a 5 AP Calculus BC 2019

AP-Calculus-BC 2727-MA-Book May 11, 2018 14:10

74 STEP 4. Review the Knowledge You Need to Score High

25. To find horizontal asymptotes, examine
    the limx→∞f(x) and the limx→−∞f(x). The

xlim→∞f(x)=xlim→∞

x

√

x^2 + 4

. Dividing by
the highest power ofx(and in this case,
it’sx), you obtain limx→∞
x/x
√
x^2 + 4 /x

.As

x→∞,x=

√

x^2. Thus, you have

xlim→∞

x/x

√

x^2 + 4 /

√

x^2

=xlim→∞

1

√

x^2 + 4

x^2

=xlim→∞

1

√

1 +

4

x^2

=1. Thus, the liney= 1

is a horizontal asymptote.

The limx→−∞f(x)=x→lim−∞

x

√

x^2 + 4

.

Asx→−∞,x=−

√

x^2. Thus, limx→−∞

x

√

x^2 + 4

=xlim→−∞

x/x

√

x^2 + 4 /

(

−

√

x^2

)=xlim→−∞

1

−

√

1 +

4

x^2

=−1.

Therefore, the liney=−1 is a horizontal

asymptote. As for vertical asymptotes,

f(x) is continuous and defined for all real

numbers. Thus, there is no vertical

asymptote.