2.3 Evaluating Limits Analytically
85
____ 11.
Let
and
. Find the limits.
a.
b.
c. –
d.
e.
____ 12. Suppose that and . Find the following limit.
a.
–10
–3
–23
____ 13. Suppose that and . Find the following limit.
10
–5
–25
–15
150
____ 14. Suppose that and . Find the following limit.
21
–21
86
Chapter 2: Limits and Their Properties
____
15.
Suppose that
and
. Find the following limit.
–11
–8
33
–14
____ 16. Suppose that . Find the following limit.
2
125
8
0
15
____ 17. Suppose that . Find the following limit.
–5
15
–15
3
____ 18. Find the following limit (if it exists). Write a simpler function that agrees with the given
function at all but one point.
40
–24
24
–40
2.3 Evaluating Limits Analytically
87
____ 19. Find the limit (if it exists).
a.
b.
–32
–8
____ 20. Find the limit (if it exists).
a. 6
b.
c.
d.
e. .
____ 21. Find the limit (if it exists).
a.
b.
c.
d.
e.
88 Chapter 2: Limits and Their Properties
____ 22. Determine the limit (if it exists).
6
48
10
24
____ 23. Determine the limit (if it exists).
a. 8
b.
c.
2
____ 24. Determine the limit (if it exists).
1
0
d.
e.
____ 25. Find where .
1
4
–3
0
2.3 Evaluating Limits Analytically
89
2.3 Evaluating Limits Analytically
Answer Section
90
Chapter 2: Limits and Their Properties
24.
2.4 Continuity and One-Sided Limits
91
2.4 Continuity and One-Sided Limits
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1.
Use the graph as shown to determine the following limits, and discuss the continuity
of the function at
.
(i)
(ii)
(iii)
2 , 2 , 2 ,
2 , 2 , 2 ,
92 Chapter 2: Limits and Their Properties
____ 2.
Use the graph as shown to determine the following limits, and discuss the continuity
of the function at
.
(i)
(ii)
(iii)
a.
b.
c.
d.
e.
2.4 Continuity and One-Sided Limits
93
____ 3.
Use the graph to determine the following limits, and discuss the continuity of the
function at
.
(i)
(ii)
(iii)
a.
b.
c.
d.
e.
____ 4. Find the limit (if it exists).
a.
b.
c.
d.
e.
94 Chapter 2: Limits and Their Properties
____ 5. Find the limit (if it exists).
a.
b.
c.
d.
e.
____ 6. Find the limit (if it exists).
a.
b.
c.
d.
e.
____ 7. Find the limit (if it exists). Note that represents the greatest integer
function.
a.
b.
c.
d.
e.
2.4 Continuity and One-Sided Limits
95
____
8.
Find the limit (if it exists). Note that
represents the greatest integer
function.
a.
b.
c.
d.
e.
____
9.
Discuss the continuity of the function
.
a.
b.
c.
d.
e.
____ 10. Find the x-values (if any) at which the function is not continuous. Which of the
discontinuities are removable?
a.
b.
c.
d.
e.
96
Chapter 2: Limits and Their Properties
____
11.
Find the x-values (if any) at which
is not continuous.
a.
b.
c.
d.
e.
____
12.
Find the x-values (if any) at which the function
is not continuous.
Which of the discontinuities are removable?
10 and – , removable
discontinuous everywhere
continuous everywhere
10 and – , not removable
0, removable
____
13.
Find the x-values (if any) at which the function
is not continuous.
Which of the discontinuities are removable?
a.
b.
c.
d.
e.
____
14.
Find the x-values (if any) at which
is not continuous.
a.
b.
c.
d.
e.