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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.2 • Limits Section of Functions 1.1 • Assess UsingYour Properties Understanding of Limits 89 95
(b) Investigate lim cos π by using a table and THEOREM evaluating the Limit of a Root (c) Graph the function C.
x→0 x2 function f (x) = cos π x 2 at
If lim f (x) exists and if n (d) ≥ 2Use is antheinteger, graph tothen
investigate lim C(w) and lim C(w). Do
x→c w→1− w→1 +
x =− 2 3 , − 2 5 , − 2 7 , − 2 9 ,..., 2 9 , 2 7 , 2 5 , 2 these √suggest that √ lim C(w) exists?
3 .
n
lim f (x) = n w→1 lim f (x)
(e) x→c Use the graph to investigate x→c lim C(w) and lim C(w).
w→12− w→12
(c) Compare the results from (a) and (b). What do you conclude
+ provided f (x) >0 if n is even. Do these suggest that lim C(w) exists?
about the limit? Why do you think this happens? What is your
w→12
view about using a table to draw a conclusion about limits? (f) Use the graph to investigate lim C(w).
w→0 +
(d) Use technology to graph f . Begin with the EXAMPLE x-window 8 Finding the (g) Limit Use theof grapha Root to investigate lim C(w).
w→13
[−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:
Solution
Be sure your
calculator is set to the radian mode.)
PAGE
x − 8
3
85 57. (a) Use a table to investigate lim .
lim
x→2
x→4
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 integers m and n, then
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
yweighing more than 3.5 ounces.
30Source: U.S. Postal Service Notice 123 EXAMPLE 9
Source: Submitted by the students of Millikin University.
The Limit of a Power
62.
and
The
the
definition
Limit
of
of
the
a Root
slope
are
of the
used
tangent
together
line to
to
the
find
graph
the limit
of
of
a function with a rational exponent.
f (x) − f (c)
y = f (x) at the point (c, f (c)) is m tan = lim
.
x→c x − c
THEOREM Limit of a Fractional
Another way
Power
to express
[ f (x)]
this m/n
slope is to define a new variable
If f is a function for which h = limx −f c. (x) Rewrite existsthe and slope if [ off (x)] the tangent m/n is line defined m tan using for positive h and c.
x→c
63. If f (2) = 6, can you conclude anything about lim f (x)? Explain
x→2
your reasoning. [ ] m/n
64. If lim lim [ f f = 6, can you conclude anything about f (2)? Explain
x→c (x)]m/n = lim f (x)
x→2 x→c
your reasoning.
65. The graph of f (x) = x − 3 is a straight line with a point punched
Finding the 3 − x
out. Limit of a Fractional Power [f(x)] m/n
27
(8, 27)
(a) Find a function C that models the first-class Findpostage lim(x charged, + 1) 3/2 .
x→8 (a) What straight line and what point?
20 in dollars, for a letter weighing w ounces. Assume w>0.
(b) Use the graph of f to investigate the one-sided limits of f as
(b) What is the domain of C?
Solution Let f (x) = x + 1. Near 8, x + 1 > 0, so (x + 1) 3/2 is defined. Then
x approaches 3.
[ ]
(c) Graph the function C.
3/2
10
lim
(d) Use the graph to investigate lim C(w) and lim
[ C(w). f (c) Does the graph suggest that lim f (x) exists? If so, what is it?
Do
x→3
x→8 (x)]3/2 = lim(x + 1) 3/2 = lim (x + 1) = [8 + 1] 3/2 = 9 3/2 = 27
x→8 ↑[
x→8
w→2− w→2 + m/n
66. lim [(a) f ( x)] Use m/n a=
table limtof ( investigate x)]
lim(1 + x)
these suggest that lim C(w) exists?
1/x .
■
x→c
x→c
x→0
5 w→2 8 10 x
(e) Use the graph to investigate lim C(w).
(b) Use graphing technology to graph g(x) = (1 + x) 1/x .
w→0 + See Figure 20.
Figure 20 f (x) = (x + 1) 3/2 (c) What do (a) and (b) suggest about lim(1 + x) 1/x ?
(f) Use the graph to investigate lim C(w).
x→0
w→3.5 − CAS (d) Find NOW lim(1 WORK + x) 1/x Problem . 23 and AP® Practice Problem 8.
x→0
60. First-Class Mail As of April 2016, the U.S. Postal Service
charged $0.94 postage for first-class large envelope 3 Find weighing theup Limit to of a Polynomial
and including 1 ounce, plus a flat fee of $0.21Sometimes for each additional lim f (x) can be found by substituting c for x in f (x). For example,
or partial ounce up to and including 13 ounces. First-class x→c
rates do Challenge Problems
not apply to large envelopes weighing more than 13 ounces. Forlim
Problems (5x 2 ) = 67–70, 5 liminvestigate x 2 = 5 · 2each 2 = of 20the following limits.
x→2 x→2
Source: U.S. Postal Service Notice 123
{
Since lim x n = c n if n is a positive integer, 1 if x is an integer
f (x) we = can use the Limit of a Constant Times a
(a) Find a function C that models the first-class postage x→c charged,
0 if x is not an integer
in dollars, for a large envelope weighingFunction w ounces. to Assume obtain a formula for the limit of a monomial f (x) = ax n .
w>0.
67. lim f (x) 68. lim f (x) 69. lim f (x) 70. lim f (x)
x→2 lim x→1/2 x→3 x→0
(b) What is the domain of C?
x→c (axn ) = ac n
Kathryn Sidenstricker /Dreamstime.com
Find lim
√ 3
x 2 + 11.
x→4
where a is any number.
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
√ √ √
x
2
+
seven
11 =
hours 3 lim
one
(x 2 studies,
+ 11)
the
=
closer 3 lim
to
x 2 100%
+ lim
the
11
student scores
on the ↑ final. x→4 He claims that ↑ studying x→4 significantly x→4 less than seven
Limit hours of amay Root cause one Limit to be ofunderprepared a Sum for the test, while
studying significantly more than seven hours may cause
“burnout.” = 3√ 4 2 + 11 = 3√ 27 = 3
↑
(a) Write Professor Smith’s claim symbolically as a limit.
lim x 2 = c 2
■
x→c (b) Write Professor Smith’s claim using the ε-δ definition
NOWofWORK limit. Problem 19 and AP® Practice Problems 6 and 7.
Section 1.2 • Limits of Functions Using Properties of Limits
95
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