5 Steps to a 5 AP Calculus BC 2019

Limits and Continuity 57

5.2 Limits Involving Infinities

Main Concepts:Infinite Limits (asx→a), Limits at Infinity (asx →∞), Horizontal

and Vertical Asymptotes

Infinite Limits (asx→a)

If f is a function defined at every number in some open interval containinga, except

possibly ataitself, then

(1) limx→a f(x)=∞means that f(x) increases without bound asxapproachesa.

(2) limx→a f(x)=−∞means that f(x) decreases without bound asxapproachesa.

Limit Theorems

(1) Ifnis a positive integer, then

(a) xlim→ 0 +

1

xn

=∞

(b) xlim→ 0 −

1

xn

=

{

∞ ifnis even

−∞ ifnis odd

.

(2) If the limx→a f(x)=c,c>0, and limx→ag(x)=0, then

limx→a

f(x)

g(x)

=

{

∞ ifg(x) approaches 0 through positive values

−∞ ifg(x) approaches 0 through negative values

.

(3) If the limx→a f(x)=c,c<0, and limx→ag(x)=0, then

limx→a

f(x)

g(x)

=

{

−∞ ifg(x) approaches 0 through positive values

∞ ifg(x) approaches 0 through negative values

.

(Note that limit theorems 2 and 3 hold true forx→a+andx→a−.)

Example 1

Evaluate the limit: (a) limx→ 2 +

3 x− 1

x− 2

and (b) limx→ 2 −

3 x− 1

x− 2

.

The limit of the numerator is 5 and the limit of the denominator is 0 through positive

values. Thus, limx→ 2 +

3 x− 1

x− 2

=∞. (b) The limit of the numerator is 5 and the limit of the

denominator is 0 through negative values. Therefore, limx→ 2 −

3 x− 1

x− 2

=−∞. Verify your result

with a calculator. (See Figure 5.2-1.)