Limits and Continuity 69
5.5 Practice Problems
Part A The use of a calculator is not allowed.
Find the limits of the following:
- limx→ 0 (x−5) cosx
- Ifb=/0, evaluate limx→b
x^3 −b^3
x^6 −b^6
.
- limx→ 0
2 −
√
4 −x
x
- xlim→∞
5 − 6 x
2 x+ 11 - xlim→−∞
x^2 + 2 x− 3
x^3 + 2 x^2 - xlim→∞
3 x^2
5 x+ 8 - xlim→−∞
3 x
√
x^2 − 4 - If f(x)=
{
ex for 0≤x< 1
x^2 ex for 1≤x≤ 5
,
find limx→ 1 f(x).
- xlim→∞
ex
1 −x^3 - limx→ 0
sin 3x
sin 4x - xlim→ 3 +
√
t^2 − 9
t− 3
- The graph of a functionf is shown in
Figure 5.5-1.
Which of the following statements is/are
true?
I.xlim→ 4 −f(x)= 5.
II.xlim→ 4 f(x)= 2.
III. x=4 is not in the domain of f.
Part B Calculators are allowed.
- Find the horizontal and vertical asymptotes
of the graph of the function
f(x)=
1
x^2 +x− 2
. 8 7 6 5 4 3 2 1
0123456789
y
x
f
Figure 5.5-1
- Find the limit: limx→ 5 +
5 +[x]
5 −x
when [x] is the
greatest integer ofx.
- Find allx-values where the function
f(x)=
x+ 1
x^2 + 4 x− 12
is discontinuous.
- For what value ofkis the function
g(x)=
{
x^2 +5, x≤ 3
2 x−k, x> 3
continuous at
x=3?
- 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.