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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 93
(b) Investigate lim cos π by using a table and evaluating the
x→0 x2 COROLLARY Limit of a Constant (c) GraphTimes the function a Function C.
function f (x) = cos π x 2 at
If g is a function for which(d) lim Use g(x) the exists graphand to investigate if k is any real lim number, C(w) and then lim limC(w). [kg(x)] Do
x→c w→1− w→1 x→c +
exists and
x =− 2 3 , − 2 5 , − 2 7 , − 2 9 ,..., 2 9 , 2 7 , 2 5 , 2 these suggest that lim C(w) exists?
IN WORDS The limit of a constant
3 .
w→1
times a function equals the constant
(e) lim Use[kg(x)] the graph= tok investigate lim g(x) lim C(w) and
x→c x→c lim C(w).
w→12− w→12
times (c) theCompare limit of the function. results from (a) and (b). What do you conclude
+ Do these suggest that lim C(w) exists?
about the limit? Why do you think this happens? You areWhat askedisto your prove this corollary in Problem 103. w→12
view about using a table to draw a conclusion about limits? (f) Use the graph to investigate lim
Limit properties often are used in combination.
C(w).
w→0 +
(d) Use technology to graph f . Begin with the x-window
(g) Use the graph 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: Find: Be sure your
calculator is set to the radian mode.)
PAGE
x − 8 (a) lim
85 57. (a) Use a table to investigate lim .
x→2 2
(b) How close must x be to 2, so that f (x) isSolution within 0.1(a)
of the
limit?
lim
(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?
NOTE (c) The Howlimit close properties must x beare toalso 2, sotrue
that f (x) is within 0.01 of the
(b) We use properties of limits Source: to find Submitted the one-sided by the students limit. of Millikin University.
for one-sided limit? limits.
62. The definition of the slope of [ the tangent ][ line to the graph ] of
lim
f (x) − f (c)
59. First-Class Mail As of April
x→2 +[4x(2
− x)] = 4 lim
y = x→2 f (x) +[x(2 at the − point x)] (c, = 4 lim
f (c)) is m x lim
x→2 + tan = x→2 lim +(2 − x) .
2016, the U.S. Postal Service
[
]
x→c x − c
charged $0.47 postage for
Another = 4 · 2 way limto 2 express − lim this x slope = 4 is · 2 to· define (2 − 2) a new = 0variable
■
x→2 + x→2 +
first-class letters weighing up to
h = x − c. Rewrite the slope of the tangent line m tan using h and c.
and including 1 ounce, plus a flat
63. If f (2) = 6, can you conclude anything NOW about WORK lim f (x)? Problem Explain 13.
fee of $0.21 for each additional
x→2
your reasoning.
or partial ounce up to and
including 3.5 ounces. First-class
To find the limit of 64. piecewise-defined If lim f (x) = 6, can x→2
functions you conclude at numbers anythingwhere about f the (2)? defining Explain
letter rates do not apply to letters equation changes requires the your use reasoning. of one-sided limits.
RECALL weighing The more limit Lthan of a3.5 function ounces.
65. The graph of f (x) = x − 3 is a straight line with a point punched
y = Source: f (x) asU.S. x approaches Postal Service a Notice 123
3 − x
EXAMPLE 5 Finding aout.
Limit for a Piecewise-defined Function
number c exists if and only if
lim
(a)
f (x)
Find
=
a
lim
function C that models the first-class postage charged,
in dollars, for f (x) a letter = L. Find lim f (x), if it exists. (a) What straight line and what point?
x→c− x→2
x→c + 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?
3x + 1 if x < 2
f (x) = x approaches 3.
(c) Graph the function C.
(c) Does
2x(x
the graph
− 1)
suggest
if
that
x
lim
≥ 2
f (x) exists? If so, what is it?
(d) Use the graph to investigate lim C(w) and lim C(w). Do
x→3
w→2− w→2 +
Solution Since the rule66. for(a) f changes Use a table at 2, towe investigate need tolim
find(1 the + x) one-sided
these suggest that lim C(w) exists?
1/x . limits of f as
x→0
w→2 x approaches 2.
(e) Use the graph to investigate lim C(w).
(b) Use graphing technology to graph g(x) = (1 + x) 1/x .
w→0 + For x < 2, we use the left-hand
(c) What
limit.
do
Also,
(a) and
because
(b) suggest
x <
about
2, f (x)
lim
=
(1
3x
+ x)
+ 1/x 1.
?
y
(f) Use the graph to investigate lim C(w).
x→0
20
w→3.5 − lim f (x) = lim CAS (d) Find lim(1 x) 1/x .
x→2 −
x→2−(3x + 1) = lim
x→2 x→0 −(3x)
+ lim 1 = 3 lim x + 1 = 3(2) + 1 = 7
x→2 − x→2
60. First-Class Mail 16
− As of April 2016, the U.S. Postal Service
charged $0.94 12 postage for first-class large envelope
For x
weighing
≥ 2, we
up
use
to
the right-hand limit. Also, because x ≥ 2, f (x) = 2x(x − 1).
and including 1 ounce, plus a flat fee of $0.21 for each additional
8
or partial ounce up to and including 13 ounces. First-class rates do Challenge Problems
not apply to large 4 envelopes weighing more than 13 ounces. lim f (x) = lim
x→2 + Forx→2 Problems +[2x(x
− 1)] = lim
67–70, investigate x→2 +(2x)
· lim
eachx→2 of the +(x
− 1)
(2, 4)
following limits.
Source: U.S. Postal Service Notice 123
[ { ] 1 if x is an integer
4 2
2 4 x
= 2 lim f (x) =
(a) Find a function C that models the first-class postage charged,
x · lim x − lim 1 = 2 · 2 [2 − 1] = 4
x→2 + x→2 + x→2 0 + if x is not an integer
in dollars, for a large envelope weighing w ounces. Assume
w>0.
Since lim f (x) = 7 = 67. lim limf (x) f (x) = 4, 68. lim lim f (x) f does (x) not 69. exist. lim ■f (x) 70. lim
{ f (x)
x→2 −
x→2 x→2 +
x→2x→1/2 x→3 x→0
(b) What is the domain 3x + 1 of C? if x < 2
Figure 19 f (x) =
2x(x − 1) if x ≥ 2 See Figure 19.
NOW WORK Problem 73 and AP® Practice Problems 1 and 5.
Kathryn Sidenstricker /Dreamstime.com
EXAMPLE 4
Finding a Limit
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
[2x(x + 4)] seven (b) hours limone studies, the closer to 100% the student scores
x→1 x→2 +[4x(2
− x)]
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
lim“burnout.”
lim
Limit of a Product
[ ][ ]
[(2x)(x + 4)] = (2x) (x + 4)
x→1 x→1 x→1
[ ] [
]
= 2 ·(a) limWrite x · Professor lim x + Smith’s lim 4 claimLimit symbolically of a Constant as a limit. Times a
(b) x→1 x→1 x→1 Write Professor Smith’s claim
Function,
using the
Limit
ε-δ definition
of a Sum
= (2 · 1) · of (1 limit. + 4) = 10 Use (2) and (1), and simplify.
AP® Exam Tip
If you like to use released AP ® Calculus
multiple-choice questions, be sure to
review each question carefully, as many
questions about limits require more
knowledge than students will have
at this point in the term. Once they
complete this chapter, students will be
able to find limits by direct substitution
or with some algebraic manipulation
followed by substitution.
AP® Exam Tip
Starting with the 2017 exam, L’Hôpital’s
Rule is on the Calculus AB exam.
L’Hôpital’s Rule is a powerful technique
that can be used to find limits of
expressions of the form 0 0 or ∞ ∞ . This
topic will be studied in Section 4.5.
Alternate Example
Finding a Limit for a Piecewise-Defined
Function
⎧ x
Given
⎪ | + 1|
fx ( ) =
x ≤ 0
⎨
2
⎩⎪ 2−
2x
x > 0
.
Find
lim fx ( ) if it exists.
x→0
Solution
Because the rule for f changes at 0, we find
the one-sided limits of f as x approaches
0. For values of x near 0 and less than 0,
f (x) = |x + 1| = x + 1 and
lim fx ( ) = lim x + 1=
1
−
−
x→0 x→0
For values of x near 0 and greater than
zero, f (x) = 2 – 2x 2 and
2
lim fx ( ) = lim 2− 2x
= lim 2− 2(0) = 2
+ + +
x→0 x→0
x→0
Because
lim fx ( )
x→0
lim fx ( ) ≠ lim fx ( ),
x→0
− x→ 0
+
does not exist.
does not exist.
Section 1.2 • Limits of Functions Using Properties of Limits
93
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