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

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

Limits and Continuity 103

8. Find the vertical and horizontal asymptotes off(x)=

1

x^2 − 25

.

Answer: The vertical asymptotes arex=±5, and the horizontal asymptote isy=0,

since lim

x→±∞

f(x)=0.

6.5 Practice Problems

Part A The use of a calculator is not allowed.

Find the limits of the following:

1. limx→ 0 (x−5) cosx


2. Ifb=0, evaluate lim
   x→b

x^3 −b^3

x^6 −b^6

.

3. limx→ 0

2 −

√

4 −x

x

4. limx→∞
   5 − 6 x
   2 x+ 11


5. limx→−∞
   x^2 + 2 x− 3
   x^3 + 2 x^2


6. limx→∞
   3 x^2
   5 x+ 8


7. limx→−∞
   3 x
   √
   x^2 − 4


8. Iff(x)=

{

ex for 0≤x< 1

x^2 ex for 1 ≤x≤ 5

,

find limx→ 1 f(x).

9. limx→∞
   ex
   1 −x^3


10. limx→ 0
    sin 3x
    sin 4x


11. limx→ 3 +

√

t^2 − 9

t− 3

12. The graph of a functionf is shown in
    Figure 6.5-1.
    Which of the following statements is/are
    true?
    I. limx→ 4 −f(x)= 5.
    II. lim
    x→ 4
    f(x)= 2.
    III. x=4 is not in the domain of f.

8 7 6 5 4 3 2 1

0123456789

y

x

f

Figure 6.5-1

Part B Calculators are allowed.

13. Find the horizontal and vertical asymptotes
    of the graph of the function
    f(x)=

1

x^2 +x− 2

.

14. Find the limit: lim
    x→ 5 +

5 +[x]

5 −x

when [x]isthe

greatest integer ofx.

15. Find allx-values where the function
    f(x)=
    x+ 1
    x^2 + 4 x− 12
    is discontinuous.


16. For what value ofkis the function

g(x)=

{

x^2 +5, x≤ 3

2 x−k, x> 3

continuous at

x=3?

17. Determine if

f(x)=

⎧

⎨

⎩

x^2 + 5 x− 14

x− 2

,ifx= 2

12, ifx= 2

is continuous atx=2. Explain why or why not.