Sullivan Microsite DigiSample

Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
154 Chapter 1 • Limits and Continuity
TRM Full Solutions to Chapter 1
Review Exercises and AP® Review
Problems
Answers to Chapter 1 Review
Exercises
− x
1. lim 1 cos = 0. For table,
x→0
1+
cos x
see TSM.
2. lim fx ( ) =−3.
3.
x→1
y
22 21
22
24
26
28
1 2 x
lim fx ( ) does not exist.
x→2
y
7
6
5
4
3
2
4. (a) 3
(b) 2
5. −
x
3
2
1
6. 6x + 2
7. 1
8. 7 2
9. −π
10. 2
11. 0
12. 0
13. 27
14. 3
15. 4
16. 1 2
17. 2 3
18. −1
19. − 1 6
20. − 1 4
1 2 3 4
x
21. 6
22. 0
23. 1
24. 1
25. −1
26. 3
27. 8
28. 0
OBJECTIVES
AP® Review
Section You should be able to . . . Example Review Exercises Problems
1.1 1 Discuss the slope of a tangent line to a graph (p. 78) 1 4
2 Investigate a limit using a table (p. 80) 2–4 1 5
3 Investigate a limit using a graph (p. 82) 5–8 2, 3
1.2 1 Find the limit of a sum, a difference, and a product (p. 91) 1–6 8, 10, 12, 14, 22,
26, 29, 30, 47, 48
2 Find the limit of a power and the limit of a root (p. 94) 7–9 11, 18, 28, 55
3 Find the limit of a polynomial (p. 95) 10 10, 22
4 Find the limit of a quotient (p. 96) 11–14 13–17, 19–21, 3
23–25, 27, 56
5 Find the limit of an average rate of change (p. 98) 15 37
6 Find the limit of a difference quotient (p. 99) 16 5, 6, 49
1.3 1 Determine whether a function is continuous at a number (p. 103) 1–4 31–36 11
2 Determine intervals on which a function is continuous (p. 106) 5, 6 39–42 8
3 Use properties of continuity (p. 108) 7, 8 39–42
4 Use the Intermediate Value Theorem (p. 110) 9, 10 38, 44–46 6
1.4 1 Use the Squeeze Theorem to find a limit (p. 117) 1 7, 69
2 Find limits involving trigonometric functions (p. 119) 2, 3 9, 51–55 4, 10
3 Determine where the trigonometric functions are continuous (p. 122) 4 63–65
4 Determine where an exponential or a logarithmic 5 43
function is continuous (p. 124)
1.5 1 Investigate infinite limits (p. 128) 1–3 57, 58
2 Find the vertical asymptotes of a graph (p. 131) 4 61, 62 1
3 Investigate limits at infinity (p. 131) 5–10 59, 60 2
4 Find the horizontal asymptotes of a graph (p. 137) 11 61, 62 1
5 Find the asymptotes of the graph of a rational function (p. 138) 12 67, 68 7
1.6 1 Use the ε-δ definition of a limit (p. 145) 1–7 50, 66
REVIEW EXERCISES
1 − cos x
1. Use a table of numbers to investigate lim x→0 1 + cos x .
In Problems 2 and 3, use a graph to investigate lim f (x).
{ x→c
2x − 5 if x < 1
2. f (x) =
at c = 1
6 − 9x if x ≥ 1
{
x
3. f (x) =
2 + 2 if x < 2
at c = 2
2x + 1 if x ≥ 2
4. For f (x) = x 2 − 3:
(a) Find the slope of the secant line joining (1, −2) and (2, 1).
(b) Find the slope of the tangent line to the graph
of f at (1, −2).
In Problems 5 and 6, for each function find the limit of the difference
f (x + h) − f (x)
quotient lim
.
h→0 h
5. f (x) = 3 6. f (x) = 3x 2 + 2x
x
7. Find lim f (x) if 1 + sin x ≤ f (x) ≤|x|+1
x→0
In Problems 8–22, find each limit.
8. lim
(2x − 1 )
x→2 x
9. lim x→π
(x cos x)
29. lim fx ( ) = 7; lim fx ( ) = 7; lim fx ( ) = 7.
x→2
− x→ 2
+ x→2
(
10. lim x 3 + 3x 2 − x − 1 ) √ 3
11. lim x(x + 2) 3
x→−1
x→0
12. lim [(2x + 3)(x 5 x 3 − 27
+ 5x)] 13. lim
x→0 x→3 x − 3
( x 2
14. lim x→3 x − 3 − 3x )
x 2 − 4
15. lim
x − 3
x→2 x − 2
x 2 + 3x + 2
x 3 + 5x 2 + 6x
16. lim
x→−1 x 2 17. lim
+ 4x + 3
x→−2 x 2 + x − 2
18. lim
(x 2 − 3x + 1 ) 15
√
3 − x
19. lim
2 + 5
x→1 x
x→2 x 2 − 4
20. lim x→0
{ 1
x
[
1
(2 + x) 2 − 1 4
]}
21. lim x→0
(x + 3) 2 − 9
x
22. lim[(x 3 − 3x 2 + 3x − 1)(x + 1) 2 ]
x→1
In Problems 23–28, find each one-sided limit, if it exists.
x 2 + 5x + 6
23. lim
x→−2 + x + 2
|x − 5|
24. lim
x→5 + x − 5
x 2 − 16
26. lim
27. lim
x→ 3/2 +2x x→4 − x − 4
In Problems 29 and 30, find lim f (x) and lim
x→c− x→c
given c. Determine whether lim f (x) exists.
{ x→c
2x + 3 if x < 2
29. f (x) =
at c = 2
9 − x if x ≥ 2
|x − 1|
25. lim
x→1 − x − 1
28. lim
x→1 + √
x − 1
+
f (x) for the
154
Chapter 1 • Limits and Continuity
TE_Sullivan_Chapter01_PART II.indd 37
11/01/17 9:57 am