5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 7, 2018 9:52

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

6.2 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) limx→ 0 +

1

xn

=∞

(b) limx→ 0 −

1

xn

=

{

∞ ifnis even

−∞ ifnis odd

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

xlim→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

xlim→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) lim

x→ 2 +

3 x− 1

x− 2

and (b) lim

x→ 2 −

3 x− 1

x− 2

.

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

values. Thus, lim

x→ 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 6.2-1.)