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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.1 • Assess Your Understanding 89
(b) Investigate lim
cos π by using a table and evaluating the
x→0 x 2 function f (x) = cos π x 2 at
x =− 2 3 , − 2 5 , − 2 7 , − 2 9 ,..., 2 9 , 2 7 , 2 5 , 2 3 .
(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.)
PAGE
x − 8
85 57. (a) Use a table to investigate lim
.
x→2 2
(b) How close must x be to 2, so that f (x) is within 0.1 of the
limit?
(c) How close must x be to 2, so that f (x) is within 0.01 of the
limit?
58. (a) Use a table to investigate lim
(5 − 2x).
x→2 (b) How close must x be to 2, so that f (x) is within 0.1 of the
limit?
(c) How close must x be to 2, so that f (x) is within 0.01 of the
limit?
59. First-Class Mail As of April
2016, the U.S. Postal Service
charged $0.47 postage for
first-class letters weighing up to
and including 1 ounce, plus a flat
fee of $0.21 for each additional
or partial ounce up to and
including 3.5 ounces. First-class
letter rates do not apply to letters
weighing more than 3.5 ounces.
Source: U.S. Postal Service Notice 123
(a) Find a function C that models the first-class postage charged,
in dollars, for a letter weighing w ounces. Assume w>0.
(b) What is the domain of C?
(c) Graph the function C.
(d) Use the graph to investigate lim
C(w) and lim
C(w). Do
w→2 − w→2 + these suggest that lim
C(w) exists?
w→2 (e) Use the graph to investigate lim
C(w).
w→0 + (f) Use the graph to investigate
lim
C(w).
w→3.5 − 60. First-Class Mail As of April 2016, the U.S. Postal Service
charged $0.94 postage for first-class large envelope weighing up to
and including 1 ounce, plus a flat fee of $0.21 for each additional
or partial ounce up to and including 13 ounces. First-class rates do
not apply to large envelopes weighing more than 13 ounces.
Source: U.S. Postal Service Notice 123
(a) Find a function C that models the first-class postage charged,
in dollars, for a large envelope weighing w ounces. Assume
w>0.
(b) What is the domain of C?
55. (a) For table see TSM. Values in table
π
suggest lim cos = 1.
x→0
x
(b) For table see TSM. Values in table
π
suggest lim cos =− 1.
x→0
x
(c) Limit does not exist (see TSM).
(d)
y
1
0.5
23 22 21 1 2 3 x
20.5
21
y
1
0.5
Kathryn Sidenstricker /Dreamstime.com
(c) Graph the function C.
(d) Use the graph to investigate lim
C(w) and lim
C(w). Do
w→1 − w→1 + these suggest that lim
C(w) exists?
w→1 (e) Use the graph to investigate
lim
C(w) and
lim
C(w).
w→12 − w→12 + Do these suggest that lim
C(w) exists?
w→12 (f) Use the graph to investigate lim
C(w).
w→0 + (g) Use the graph to investigate
lim
C(w).
w→13 −
61. Correlating Student Success to Study Time Professor Smith
claims that a student’s final exam score is a function of the time t
(in hours) that the student studies. He claims that the closer to
seven hours one studies, the closer to 100% the student scores
on the final. He claims that studying significantly less than seven
hours may cause one to be underprepared for the test, while
studying significantly more than seven hours may cause
“burnout.”
(a) Write Professor Smith’s claim symbolically as a limit.
(b) Write Professor Smith’s claim using the ε-δ definition
of limit.
Source: Submitted by the students of Millikin University.
62. The definition of the slope of the tangent line to the graph of
f (x) − f (c)
y = f (x) at the point (c, f (c)) is m tan = lim
.
x→c x − c
Another way to express this slope is to define a new variable
h = x − c. Rewrite the slope of the tangent line m tan using h and c.
63. If f (2) = 6, can you conclude anything about lim
f (x)? Explain
x→2 your reasoning.
64. If lim
f (x) = 6, can you conclude anything about f (2)? Explain
x→2 your reasoning.
65. The graph of f (x) = x − 3
is a straight line with a point punched
3 − x out.
(a) What straight line and what point?
(b) Use the graph of f to investigate the one-sided limits of f as
x approaches 3.
(c) Does the graph suggest that lim
f (x) exists? If so, what is it?
x→3 66. (a) Use a table to investigate lim
(1 + x) 1/x .
x→0 (b) Use graphing technology to graph g(x) = (1 + x) 1/x .
(c) What do (a) and (b) suggest about lim
(1 + x) 1/x ?
x→0 CAS
(d) Find lim
(1 + x) 1/x .
x→0
Challenge Problems
For Problems 67–70, investigate each of the following limits.
{ 1 if x is an integer
f (x) =
0 if x is not an integer
67. lim
f (x) 68. lim
f (x) 69. lim
f (x) 70. lim
f (x)
x→2 x→1/2 x→3 x→0
56. (a) For table see TSM. Values in the table
π
suggest lim cos =
→ x
1 .
x 0
2
(b) For table see TSM. Values in table
π 2
suggest. lim cos = .
x→0
2
x 2
(c) Limit does not exist (see TSM).
(d)
y
1
0.5
24 23 22 21 1 2 3 4 x
20.5
21
y
1
0.5
21 20.5 0.5 1 x
20.5
21
x − 8
57. (a) lim =− 3. For sample table
x→2
2
see TSM.
(b) 1.8≤ x ≤2.2
(c) 1.98 ≤ x ≤2.02
58. (a) lim (5− 2 x) = 1. For sample
x→ 2
table see TSM.
(b) 1.95 ≤ x ≤2.05
(c) 1.995 ≤ x ≤2.005
⎧0.47 if 0< w ≤1
⎪
⎪0.68 if 1< w ≤2
59. (a) Cw ( ) = ⎨
⎪0.89 if 2 < w ≤3
⎪
⎩⎪
1.10 if 3 < w ≤3.5
(b) { w|0< w ≤3.5}
(c)
C(w)
1.2
1.0
0.8
0.6
0.4
0.2
1 2 3 3.5 w
(d) lim Cw ( ) = 0.68, lim Cw ( ) = 0.89,
w→2
− +
w→2 w→2
lim Cw ( )does not exist.
(e)
(f)
60. (a)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
Cw ( ) = ⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
lim Cw ( ) = 0.47
w → 0
+
lim Cw ( ) = 1.10
w →3.5
−
0.94 if 0 < w ≤1
1.15 if 1< w ≤2
1.36 if 2 < w ≤3
1.57 if 3 < w ≤4
1.78 if 4 < w ≤5
1.99 if 5 < w ≤6
2.20 if 6 < w ≤7
2.41 if 7 < w ≤8
2.62 if 8 < w ≤9
2.83 if 9 < w ≤10
3.04 if 10< w ≤11
3.25 if 11< w ≤12
3.46 if 12< w ≤13
20.1 20.05 0.05 0.1 x
20.5
21 Answers continue on p. 90
Section 1.1 • Assess Your Understanding
89
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