Sullivan Microsite DigiSample

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 97
(b) Investigate lim cos π by using a table and evaluating the
x→0 x2 EXAMPLE 11 Finding(c) theGraph Limit theofunction a Rational C. Function
function f (x) = cos π x 2 at
Find:
(d) Use the graph to investigate lim C(w) and lim C(w). Do
w→1− w→1 +
x =− 2 3 , − 2 5 , − 2 7 , − 2 9 ,..., 2 9 , 2 7 , 2 5 , 2 3x 3 − 2x + 1 these suggest that 2x + lim
3 .
w→1
4 C(w) exists?
(a) lim
(b) lim
x→1 4x 2 + 5 (e) Use thex→−2
graph3x to 2 investigate − 1 lim C(w) and lim C(w).
w→12− w→12
(c) Compare the results from (a) and (b). What do you conclude
+
about the limit? Why do you think this happens? Solution What (a) isSince your 1 is in the domain
Do these
of
suggest
the rational
that lim
w→12 function
C(w) exists?
R(x) = 3x 3 − 2x + 1
,
view about using a table to draw a conclusion about limits? (f) Use the graph to investigate lim C(w). 4x 2 + 5
w→0
lim
+
(d) Use technology to graph f . Begin with the x-window
(g) R(x) Use = R(1) = 3 − 2 + 1 = 2
x→1 ↑the graph to investigate 4 + 5 lim9
C(w).
w→13
[−2π, 2π] and the y-window [−1, 1]. If you were finding
− Use (3)
lim f (x) using a graph, what would you conclude? Zoom in 61. Correlating Student Success to Study Time Professor Smith
x→0
on the graph. Describe what you see. (Hint: (b) Be Since sure −2 youris in the domain claims of the thatrational a student’s function final exam H(x) score = is 2x a function + 4 of the time t
calculator is set to the radian mode.)
(in hours) that the student studies. He claims 3x 2 −that 1 ,
the closer to
PAGE
x − 8
seven hours one studies, the closer to 100% the student scores
85 57. (a) Use a table to investigate lim .
lim
x→2 2
on H(x) the = H(−2) = −4 + 4
x→−2 ↑final. He claims12that − studying 1 = 0
11 significantly = 0 less than seven
■
(b) How close must x be to 2, so that f (x) is within 0.1 of the hours Use may (3) cause one to be underprepared for the test, while
limit?
studying significantly more than seven hours may cause
(c) How close must x be to 2, so that f (x) is within 0.01 of the “burnout.”
NOW WORK Problem 33.
limit?
(a) Write Professor Smith’s claim symbolically as a limit.
58. (a) Use a table to investigate lim(5 − 2x).
x→2
EXAMPLE 12 Finding(b) theWrite Limit Professor a Quotient Smith’s claim using the ε-δ definition
√
(b) How close must x be to 2, so that f (x) is within 0.1 of the
of limit.
3x 2
+ 1
limit?
Find lim .
x→4 x − 1 Source: Submitted by the students of Millikin University.
(c) How close must x be to 2, so that f (x) is within 0.01 of the
62. The definition of the slope of the tangent line to graph of
limit?
Solution We seek the limit of the quotient of two functions. Since the limit of the
f (x) − f (c)
59. First-Class Mail As of April
denominator lim(x − 1) = 0, we use the Limit of a Quotient.
x→4 y = f (x) at the point (c, f (c)) is m tan = lim
.
x→c
2016, the U.S. Postal Service
x − c
√
√ √
charged $0.47 postage for
3x 2
+ 1
lim 3x Another 2
+ 1way to express lim (3x 2 √
this + slope 1)
√
is to define a new variable
x→4
x→4 3 · 42 + 1 49
first-class letters weighing up to
lim = h = x − c. Rewrite the slope of the tangent line m tan using h and c.
x→4 x − 1 lim
and including 1 ounce, plus a flat
63. (x If − f (2) 1) =
lim
= 6, can you (x conclude − 1) =
= = 7 4 − 1 3 3
↑ x→4 ↑ x→4 anything about lim f (x)? Explain
fee of $0.21 for each additional
Limit of a Quotient
x→2
your
Limit
reasoning.
of a Root
■
or partial ounce up to and
64. If lim f (x) = 6, can you conclude anything NOW WORK about f (2)? Problem Explain 31.
including 3.5 ounces. First-class
x→2
letter rates do not apply to letters
Based on these examples, your you reasoning. might be tempted
weighing more than 3.5 ounces.
65. The graph of f (x) = x − 3 to conclude that finding a limit as x
approaches c is simply a matter of substituting the number is a straight c intoline thewith function. a pointThe punched next
Source: U.S. Postal Service Notice 123 few examples show that substitution
3 − x
out. cannot always be used and other strategies need to
be employed.
(a) Find a function C that models the first-class postage charged, (a) What straight line and what point?
in dollars, for a letter weighing w ounces. Assume The w>0. limit of a rational function can be found using substitution, provided the
(b) Use the graph of f to investigate the one-sided limits of f as
(b) What is the domain of C?
number c being approached is in the domain of the rational function. The next example
x approaches 3.
(c) Graph the function C.
shows a strategy that can be tried when c is not in the domain.
(c) Does the graph suggest that lim 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 +
66. (a) Use a table to investigate lim(1 + x)
these suggest that lim C(w) exists? EXAMPLE 13 Finding the Limit of a Rational 1/x .
x→0
Function
w→2
(e) Use the graph to investigate lim
x 2 + 5x + 6 (b) Use graphing technology to graph g(x) = (1 + x) 1/x .
.
x 2 − 4 (c) What do (a) and (b) suggest about lim
Kathryn Sidenstricker /Dreamstime.com
C(w).
w→0 + Find lim
(f) Use the graph to investigate lim C(w).
x→−2
(1 + x) 1/x ?
x→0
w→3.5 − Solution Since −2 is not CAS in(d) theFind domain lim(1 of+ the x) 1/x rational . function, substitution cannot be
60. First-Class Mail As of April 2016, the U.S. used. Postal But Service this does not mean that the x→0
limit does not exist! Factoring the numerator and
charged $0.94 postage for first-class large envelope the denominator, weighing up to we find
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 Challenge x 2 + 5x Problems + 6 (x + 2)(x + 3)
=
not apply to large envelopes weighing more than 13 ounces.
x
For Problems 2 − 4 (x + 2)(x − 2)
67–70, investigate each of the following limits.
Source: U.S. Postal Service Notice 123 Since x = −2, and we are interested in the limit
{
as 1 x approaches if x is an integer −2, the factor x + 2
f (x) =
(a) Find a function C that models the first-class canpostage be divided charged, out. Then
0 if x is not an integer
in dollars, for a large envelope weighing w ounces. Assume x 2 + 5x + 6 (x + 2)(x + 3)
w>0.
lim
67. = lim
f (x) 68. lim f (x) 69. lim f (x) 70. lim f (x)
x→−2 x 2 − 4 x→−2 x→2 (x + 2)(x −x→1/2 2) = lim x + 3
x→−2 x − 2 = −2 + 3
↑
↑
x→3
↑ −2 − 2 =−1 x→0 4
(b) What is the domain of C?
Factor
x = −2
Divide out ( x + 2)
Use the Limit of a
Rational Function
NOW WORK Problem 35 and AP® Practice Problem 2.
■
AP® Calc Skill Builder
for Example 11
Finding the Limit of a Rational Function
3
x − 9x
Find lim
→ x − 3 .
x 3
Solution
3
x 9x
x x
lim lim ( 2
−
−
=
9)
x→3
x − 3 x→3
x − 3
x( x− 3)( x + 3)
= lim
x→3
x − 3
= lim x( x+ 3) = 3(3+ 3) = 18
x→3
To reinforce notational fluency, remind
students that the limit notation must be
included in each step of the factorization.
Also, encourage them to proceed gradually,
step by step, when factoring expressions.
For your visual learners, use the graph of
the function to explain this limit.
24
22
y
18
15
12
9
6
3
Alternate Example
Finding the Limit of a Rational Function
x − x
Find lim 9 3/2 7/2
.
→ + 5/2 3/2
x 0 x − 3x
Solution
Factor numerator and denominator:
x − x
lim 9
→0
x − 3x
x
3/2 7/2
=
lim
+ 5/2 3/2 +
x→0
x
x
2
4
x
(9 − x )
( x−
3)
3/2 2
3/2
TRM Section 1.2: Worksheet 2
In this two-page worksheet, graphs are provided
for 6 limit questions that each require some
form of algebraic manipulation. An additional 6
questions cover the properties of limits given a
table of values.
TRM Section 1.2: Worksheet 3
This worksheet contains 8 limit questions that can
be solved using techniques learned in this section.
Graphs are not provided for answer verification.
=
−x
lim
x
+
x→0
( x −9)
( x−
3)
( x− 3)( x+
3)
=−lim +
x→0
( x − 3)
=− lim ( x + 3) =−3
+
x→0
3/2 2
3/2
Section 1.2 • Limits of Functions Using Properties of Limits
97
TE_Sullivan_Chapter01_PART 0.indd 26
11/01/17 9:52 am