5 Steps to a 5 AP Calculus BC 2019

Limits and Continuity 67

5.4 Rapid Review

1. Findf(2) and limx→ 2 f(x) and determine iffis continuous atx=2. (See Figure 5.4-1.)

Answer: f(2)=2, limx→ 2 f(x)=4, and fis discontinuous atx=2.

20

2

4

(4, 2)

y

x

f(x)

(2, 2)

Figure 5.4-1

2. Evaluate limx→a
   x^2 −a^2
   x−a

.

Answer: limx→a

(x+a)(x−a)

x−a

= 2 a.

3. Evaluate limx→∞
   1 − 3 x^2
   x^2 + 100 x+ 99

.

Answer: The limit is−3, since the polynomials in the numerator and denominator

have the same degree.

4. Determine iff(x)=

{

x+6 forx< 3

x^2 forx≥ 3

is continuous atx=3.

Answer: The functionfis continuous, sincef(3)=9, limx→ 3 +f(x)=xlim→ 3 − f(x)=9, and

f(3)=limx→ 3 f(x).

5. If f(x)=

{

ex forx/= 0

5 forx= 0

, find limx→ 0 f(x).

Answer: limx→ 0 f(x)=1, since limx→ 0 +f(x)=xlim→ 0 − f(x)=1.

6. Evaluate limx→ 0
   sin 6x
   sin 2x

.

Answer: The limit is

6

2

=3, since limx→ 0

sinx

x

=1.