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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
88 Chapter 1 • Limits and Continuity
33. 2
34. Limit does not exist.
35. Limit does not exist.
36. Limit does not exist.
37. Answers will vary. Sample answer:
y
4
3
2
1
1 2 3 4
38. Answers will vary. Sample answer:
y
2
1
22 21 1 2 3 x
21
22
39. Answers will vary. Sample answer:
y
21
4
3
2
1
21
1 2
40. Answers will vary. Sample answer:
y
3
2
1
22 21 1 2 3 4 x
21
41. 1 42. −1
43. 0 44. 1
45. 1 46. 1
47. 0 48. 0
49. 0 50. −1
51. (a) m sec
= 15
(b) msec = 3( x+
2)
(c) lim m = 12. For sample table
x→2
see TSM.
(d)
y
30
25
20
15
10
5
sec
(2, 12)
(3, 27)
1 2 3
x
x
Tangent line
y 12x 12
Secant line
y 15x 18
52. (a) msec
= 19
2
(b) msec
= x + 2x
+ 4
(c) lim msec
= 12. For sample table
x→2
see TSM.
x
33.
2x 2 if x < 1
f (x) =
3x 2 − 1 if x > 1
at c = 1
34.
x 3 if x < −1
f (x) =
x 2 − 1
if x > −1
at c =−1
35.
x 2 if x ≤ 0
f (x) =
at c = 0
2x + 1 if x > 0
⎧
⎨
x 2 if x < 1
36. f (x) =
2 if x = 1
at c = 1
⎩
−3x + 2 if x > 1
Applications and Extensions
In Problems 37–40, sketch a graph of a function with the given
properties. Answers will vary.
37. lim
f (x) = 3;
x→2 lim
f (x) = 3;
x→3 − lim
f (x) = 1;
x→3 + f (2) = 3; f (3) = 1
38. lim
f (x) = 0;
x→−1 lim
f (x) =−2;
x→2 − lim
f (x) =−2;
x→2 + f (−1) is not defined; f (2) =−2
39. lim
f (x) = 4;
x→1 lim
f (x) =−1;
x→0 − lim
f (x) = 0;
x→0 + f (0) =−1; f (1) = 2
40. lim
f (x) = 2;
x→2 lim
f (x) = 0;
x→−1 lim
f (x) = 1;
x→1 f (−1) = 1; f (2) = 3
In Problems 41–50, use either a graph or a table to investigate
each limit.
|x − 5|
|x − 5|
41. lim
42. lim
43. lim
2x
x→5 + x − 5
x→5 − x − 5
x→ 12
2x
−
44. lim 2x 2x 2x
x→ 12
2x 45. lim
+
x→ 23
2x 46. lim
−
x→ 23
2x
+
47. lim
|x|−x 48. lim
|x|−x
x→2 + x→2 −
3 3 49. lim x−x x−x 50. lim x−x x−x
x→2 + x→2 − 51. Slope of a Tangent Line For f (x) = 3x 2 :
(a) Find the slope of the secant line containing the points (2, 12)
and (3, 27).
(b) Find the slope of the secant line containing the points (2, 12)
and (x, f (x)), x = = 2.
(c) Create a table to investigate the slope of the tangent line to the
graph of f at 2 using the result from (b).
(d) On the same set of axes, graph f , the tangent line to the graph
of f at the point (2, 12), and the secant line from (a).
52. Slope of a Tangent Line For f (x) = x 3 :
(d)
(a) Find the slope of the secant line containing the points (2, 8)
and (3, 27).
(b) Find the slope of the secant line containing the points (2, 8)
and (x, f (x)), x = = 2.
(c) Create a table to investigate the slope of the tangent line to the
graph of f at 2 using the result from (b).
(d) On the same set of axes, graph f , the tangent line to the graph
of f at the point (2, 8), and the secant line from (a).
y
40
30
(3, 27)
Secant line
20 y 5 19x 2 30
10
(2, 8)
1 2 3
Tangent line
y 5 12x 2 16
1
53. (a) msec = 2+
h for h ≠ 0
2
(b) For table see TSM.
(c) lim m = 2
h→0
sec
(d) m = 2
tan
x
53. Slope of a Tangent Line For f (x) = 1 2 x2 − 1:
(a) Find the slope m sec of the secant line containing the
points P = (2, f (2)) and Q = (2 + h, f (2 + h)).
(b) Use the result from (a) to complete the following table:
h −0.5 −0.1 −0.001 0.001 0.1 0.5
m sec
(c) Investigate the limit of the slope of the secant line found in (a)
as h → 0.
(d) What is the slope of the tangent line to the graph of f at the
point P = (2, f (2))?
(e) On the same set of axes, graph f and the tangent line to f at
P = (2, f (2)).
54. Slope of a Tangent Line For f (x) = x 2 − 1:
(a) Find the slope m sec of the secant line containing the
points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)).
(b) Use the result from (a) to complete the following table:
h −0.1 −0.01 −0.001 −0.0001 0.0001 0.001 0.01 0.1
m sec
(c) Investigate the limit of the slope of the secant line found
in (a) as h → 0.
(d) What is the slope of the tangent line to the graph of f at the
point P = (−1, f (−1))?
(e) On the same set of axes, graph f and the tangent line to f
at P = (−1, f (−1)).
PAGE
85 55. (a) Investigate lim cos π by using a table and evaluating the
x→0 x function f (x) = cos π x at
x =− 1 2 , − 1 4 , − 1 8 , − 1 10 , − 1 12 ,..., 1
12 , 1 10 , 1 8 , 1 4 , 1 2 .
(b) Investigate lim
cos π by using a table and evaluating the
x→0 x function f (x) = cos π x at
x =−1, − 1 3 , − 1 5 , − 1 7 , − 1 9 ,..., 1 9 , 1 7 , 1 5 , 1 3 , 1.
(c) Compare the results from (a) and (b). What do you conclude
about the limit? Why do you think this happens? What is
your view about using a table to draw a conclusion about
limits?
(d) Use technology to graph f . Begin with the x-window
[−2π, 2π] and the y-window [−1, 1]. If you were finding
lim
f (x) using a graph, what would you conclude? Zoom in
x→0 on the graph. Describe what you see. (Hint: Be sure your
calculator is set to the radian mode.)
56. (a) Investigate lim
cos π by using a table and evaluating the
x→0 x 2 function f (x) = cos π at x =−0.1, −0.01, −0.001,
x 2 −0.0001, 0.0001, 0.001, 0.01, 0.1.
(e)
3
2
1
1
y
1 2 3
y 2x 3
54. (a) msec =− 2+
h for h ≠ 0
(b) For table see TSM.
(c) lim msec
=−2
h→0
(d) m =−2
(e)
tan
y 5 22x 2 2
22 21
y
3
2
1
21
x
1
x
Answers continue on p. 89
88
Chapter 1 • Limits and Continuity
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