48
Chapter 1
Limits and Their Properties
1.2 Finding Limits Graphically and Numerically
Estimate a limit using a numerical or graphical approach.
Learn different ways that a limit can fail to exist.
Study and use a formal definition of limit.
An Introduction to Limits
To sketch the graph of the function
f 共x兲 lim f (x) = 3
x→1
(1, 3)
x3 1
x1
for values other than x 1, you can use standard curve-sketching techniques. At x 1,
however, it is not clear what to expect. To get an idea of the behavior of the graph of f
near x 1, you can use two sets of x-values—one set that approaches 1 from the left
and one set that approaches 1 from the right, as shown in the table.
y
3
x approaches 1 from the right.
x approaches 1 from the left.
2
f(x) =
x
0.75
0.9
0.99
0.999
1
1.001
1.01
1.1
1.25
f 共x兲
2.313
2.710
2.970
2.997
?
3.003
3.030
3.310
3.813
−1
x −1
x3
f 共x兲 approaches 3.
f 共x兲 approaches 3.
x
−2
−1
1
The limit of f 共x兲 as x approaches 1 is 3.
Figure 1.5
The graph of f is a parabola that has a gap at the point 共1, 3兲, as shown in
Figure 1.5. Although x cannot equal 1, you can move arbitrarily close to 1, and as a
result f 共x兲 moves arbitrarily close to 3. Using limit notation, you can write
lim f 共x兲 3.
This is read as “the limit of f 共x兲 as x approaches 1 is 3.”
x→1
This discussion leads to an informal definition of limit. If f 共x兲 becomes arbitrarily close
to a single number L as x approaches c from either side, then the limit of f 共x兲, as x
approaches c, is L. This limit is written as
lim f 共x兲 L.
x→c
Exploration
The discussion above gives an example of how you can estimate a limit
numerically by constructing a table and graphically by drawing a graph.
Estimate the following limit numerically by completing the table.
x2 3x 2
x→2
x2
lim
x
f 共x兲
1.75
1.9
1.99
1.999
2
2.001
2.01
2.1
2.25
?
?
?
?
?
?
?
?
?
Then use a graphing utility to estimate the limit graphically.
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1.2
Finding Limits Graphically and Numerically
49
Estimating a Limit Numerically
Evaluate the function f 共x兲 x兾共冪x 1 1兲 at several x-values near 0 and use the
results to estimate the limit
lim
x→ 0
y
x
.
冪x 1 1
The table lists the values of f 共x兲 for several x-values near 0.
Solution
f is undefined
at x = 0.
f(x) =
x approaches 0 from the left.
x
x+1−1
0.01
0.001
0.0001
0
0.0001
0.001
0.01
1.99499
1.99950
1.99995
?
2.00005
2.00050
2.00499
x
1
f 共x兲
f 共x兲 approaches 2.
x
−1
x approaches 0 from the right.
f 共x兲 approaches 2.
1
The limit of f 共x兲 as x approaches 0 is 2.
Figure 1.6
From the results shown in the table, you can estimate the limit to be 2. This limit is
reinforced by the graph of f (see Figure 1.6).
In Example 1, note that the function is undefined at x 0, and yet f (x) appears to
be approaching a limit as x approaches 0. This often happens, and it is important to
realize that the existence or nonexistence of f 共x兲 at x c has no bearing on the
existence of the limit of f 共x兲 as x approaches c.
Finding a Limit
Find the limit of f 共x兲 as x approaches 2, where
y
2
f (x) =
1, x ≠ 2
f 共x兲 0, x = 2
冦1,0,
x2
.
x2
Solution Because f 共x兲 1 for all x other than x 2, you can estimate that the limit
is 1, as shown in Figure 1.7. So, you can write
x
1
2
3
The limit of f 共x兲 as x approaches 2 is 1.
Figure 1.7
lim f 共x兲 1.
x→2
The fact that f 共2兲 0 has no bearing on the existence or value of the limit as x
approaches 2. For instance, as x approaches 2, the function
g 共x兲 冦2,
1, x 2
x2
has the same limit as f.
So far in this section, you have been estimating limits numerically and graphically.
Each of these approaches produces an estimate of the limit. In Section 1.3, you will
study analytic techniques for evaluating limits. Throughout the course, try to develop a
habit of using this three-pronged approach to problem solving.
1. Numerical approach
2. Graphical approach
3. Analytic approach
Construct a table of values.
Draw a graph by hand or using technology.
Use algebra or calculus.
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50
Chapter 1
Limits and Their Properties
Limits That Fail to Exist
In the next three examples, you will examine some limits that fail to exist.
Different Right and Left Behavior
Show that the limit lim
x→0
y
f 共x兲 1
f (x) = 1
−δ
1
δ
ⱍxⱍ.
x
In Figure 1.8 and from the definition of absolute value,
x
−1
x
Consider the graph of the function
Solution
⎪x⎪
f (x) = x
ⱍxⱍ does not exist.
ⱍxⱍ 冦x,
x, x 0
x < 0
Definition of absolute value
you can see that
ⱍxⱍ f (x) = − 1
x
冦1,1,
x > 0
.
x < 0
So, no matter how close x gets to 0, there will be both positive and negative x-values
that yield f 共x兲 1 or f 共x兲 1. Specifically, if (the lowercase Greek letter delta) is
a positive number, then for x-values satisfying the inequality 0 < x < , you can
classify the values of x 兾x as
lim f 共x兲 does not exist.
x→0
Figure 1.8
共 , 0兲
ⱍⱍ
ⱍⱍ
共0, 兲.
or
Negative x-values
yield x 兾x 1.
Positive x-values
yield x 兾x 1.
ⱍⱍ
ⱍⱍ
ⱍⱍ
Because x 兾x approaches a different number from the right side of 0 than it approaches
from the left side, the limit lim 共 x 兾x兲 does not exist.
x→0
ⱍⱍ
Unbounded Behavior
Discuss the existence of the limit lim
x→0
Solution
f 共x兲 1
x2
4
3
ⱍⱍ
x
1
lim f 共x兲 does not exist.
x→0
Figure 1.9
1
10
f 共x兲 1
> 100.
x2
Similarly, you can force f 共x兲 to be greater than 1,000,000, as shown.
1
−1
1
.
x2
0 < x <
2
−2
Consider the graph of the function
In Figure 1.9, you can see that as x approaches 0 from either the right or the left, f 共x兲
increases without bound. This means that by choosing x close enough to 0, you can
force f 共x兲 to be as large as you want. For instance, f 共x) will be greater than 100 when you
1
choose x within 10 of 0. That is,
y
f(x) =
1
.
x2
2
ⱍⱍ
0 < x <
1
1000
f 共x兲 1
> 1,000,000
x2
Because f 共x兲 does not become arbitrarily close to a single number L as x approaches 0,
you can conclude that the limit does not exist.
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1.2
Finding Limits Graphically and Numerically
51
Oscillating Behavior
See LarsonCalculus.com for an interactive version of this type of example.
1
Discuss the existence of the limit lim sin .
x→0
x
Solution Let f 共x兲 sin共1兾x兲. In Figure 1.10, you can see that as x approaches 0,
f 共x兲 oscillates between 1 and 1. So, the limit does not exist because no matter how
small you choose , it is possible to choose x1 and x2 within units of 0 such that
sin共1兾x1兲 1 and sin共1兾x2 兲 1, as shown in the table.
y
f (x) = sin
1
x
1
x
−1
x
1
sin
1
x
2
2
3
2
5
2
7
2
9
2
11
x→0
1
1
1
1
1
1
Limit does not exist.
−1
lim f 共x兲 does not exist.
x→0
Figure 1.10
Common Types of Behavior Associated with Nonexistence
of a Limit
1. f 共x兲 approaches a different number from the right side of c than it approaches
from the left side.
2. f 共x兲 increases or decreases without bound as x approaches c.
3. f 共x兲 oscillates between two fixed values as x approaches c.
There are many other interesting functions that have unusual limit behavior. An
often cited one is the Dirichlet function
f 共x兲 冦0,1,
if x is rational
.
if x is irrational
Because this function has no limit at any real number c, it is not continuous at any real
number c. You will study continuity more closely in Section 1.4.
PETER GUSTAV DIRICHLET
(1805–1859)
TECHNOLOGY PITFALL When you use a graphing utility to investigate the
behavior of a function near the x-value at which you are trying to evaluate a limit,
remember that you can’t always trust the pictures that graphing utilities draw. When
you use a graphing utility to graph the function in Example 5 over an interval
containing 0, you will most likely obtain an incorrect graph such as that shown in
Figure 1.11. The reason that a graphing utility can’t show the correct graph is that the
graph has infinitely many oscillations over any interval that contains 0.
1.2
In the early development of
calculus, the definition of a function
was much more restricted than it
is today, and “functions” such as
the Dirichlet function would not
have been considered.The modern
definition of function is attributed
to the German mathematician
Peter Gustav Dirichlet.
−0.25
0.25
−1.2
See LarsonCalculus.com to read
more of this biography.
Incorrect graph of f 共x兲 sin共1兾x兲
Figure 1.11
INTERFOTO/Alamy
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deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
52
Chapter 1
Limits and Their Properties
FOR FURTHER INFORMATION
For more on the introduction of
rigor to calculus, see “Who Gave
You the Epsilon? Cauchy and the
Origins of Rigorous Calculus”
by Judith V. Grabiner in The
American Mathematical Monthly.
To view this article, go to
MathArticles.com.
A Formal Definition of Limit
Consider again the informal definition of limit. If f 共x兲 becomes arbitrarily close to a
single number L as x approaches c from either side, then the limit of f 共x兲 as x approaches
c is L, written as
lim f 共x兲 L.
x→c
At first glance, this definition looks fairly technical. Even so, it is informal because
exact meanings have not yet been given to the two phrases
“f 共x兲 becomes arbitrarily close to L”
and
“x approaches c.”
The first person to assign mathematically rigorous meanings to these two phrases was
Augustin-Louis Cauchy. His ␧-␦ definition of limit is the standard used today.
In Figure 1.12, let (the lowercase Greek letter epsilon) represent a (small)
positive number. Then the phrase “f 共x兲 becomes arbitrarily close to L” means that f 共x兲
lies in the interval 共L , L 兲. Using absolute value, you can write this as
L+ε
L
(c, L)
ⱍ f 共x兲 Lⱍ <
L−ε
.
Similarly, the phrase “x approaches c” means that there exists a positive number such
that x lies in either the interval 共c , c兲 or the interval 共c, c 兲. This fact can be
concisely expressed by the double inequality
c+δ
c
c−δ
The - definition of the limit of f 共x兲
as x approaches c
Figure 1.12
ⱍ
ⱍ
0 < x c < .
The first inequality
ⱍ
ⱍ
0 < xc
The distance between x and c is more than 0.
expresses the fact that x c. The second inequality
ⱍx cⱍ < x is within units of c.
says that x is within a distance of c.
Definition of Limit
Let f be a function defined on an open interval containing c (except possibly
at c), and let L be a real number. The statement
lim f 共x兲 L
x→c
> 0 there exists a > 0 such that if
means that for each
ⱍ
ⱍ
0 < xc < then
ⱍ f 共x兲 Lⱍ <
.
REMARK Throughout this text, the expression
lim f 共x兲 L
x→c
implies two statements—the limit exists and the limit is L.
Some functions do not have limits as x approaches c, but those that do cannot have
two different limits as x approaches c. That is, if the limit of a function exists, then the
limit is unique (see Exercise 75).
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deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1.2
Finding Limits Graphically and Numerically
53
The next three examples should help you develop a better understanding of the
- definition of limit.
Finding a ␦ for a Given ␧
Given the limit
lim 共2x 5兲 1
x→3
find such that
ⱍ共2x 5兲 1ⱍ < 0.01
whenever
ⱍ
ⱍ
0 < x 3 < .
REMARK In Example 6,
note that 0.005 is the largest
value of that will guarantee
ⱍ共2x 5兲 1ⱍ < 0.01
whenever
ⱍ
ⱍ
0 < x 3 < .
Any smaller positive value of
would also work.
Solution In this problem, you are working with a given value of —namely,
0.01. To find an appropriate , try to establish a connection between the absolute
values
ⱍ共2x 5兲 1ⱍ
ⱍx 3ⱍ.
and
Notice that
ⱍ共2x 5兲 1ⱍ ⱍ2x 6ⱍ 2ⱍx 3ⱍ.
Because the inequality ⱍ共2x 5兲 1ⱍ < 0.01 is equivalent to 2ⱍx 3ⱍ < 0.01,
you can choose
12共0.01兲 0.005.
This choice works because
ⱍ
ⱍ
0 < x 3 < 0.005
implies that
ⱍ共2x 5兲 1ⱍ 2ⱍx 3ⱍ < 2共0.005兲 0.01.
As you can see in Figure 1.13, for x-values within 0.005 of 3 共x 3兲, the values of f 共x兲
are within 0.01 of 1.
1.01
1
0.99
y
2.995
3
3.005
2
1
x
1
2
3
4
−1
−2
f (x) = 2x − 5
The limit of f 共x兲 as x approaches 3 is 1.
Figure 1.13
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54
Chapter 1
Limits and Their Properties
In Example 6, you found a -value for a given . This does not prove the
existence of the limit. To do that, you must prove that you can find a for any , as
shown in the next example.
Using the ␧-␦ Definition of Limit
Use the - definition of limit to prove that
4+ε
lim 共3x 2兲 4.
x→2
4
Solution
4−ε
ⱍ共3x 2兲 4ⱍ <
2+δ
2
2−δ
y
> 0, there exists a > 0 such that
You must show that for each
whenever
ⱍ
ⱍ
0 < x 2 < .
Because your choice of depends on , you need to establish a connection between the
absolute values 共3x 2兲 4 and x 2 .
4
ⱍ
ⱍ ⱍ
ⱍ共3x 2兲 4ⱍ ⱍ3x 6ⱍ 3ⱍx 2ⱍ
3
> 0, you can choose 兾3. This choice works because
So, for a given
2
1
ⱍ
ⱍ
ⱍ
0 < x2 < f (x) = 3x − 2
3
implies that
x
1
2
3
4
The limit of f 共x兲 as x approaches 2 is 4.
Figure 1.14
ⱍ共3x 2兲 4ⱍ 3ⱍx 2ⱍ < 3冢3冣 .
As you can see in Figure 1.14, for x-values within of 2 共x 2兲, the values of f 共x兲 are
within of 4.
Using the ␧-␦ Definition of Limit
Use the - definition of limit to prove that
lim x 2 4.
x→2
Solution
f (x) = x 2
You must show that for each
> 0, there exists a > 0 such that
ⱍx 2 4ⱍ <
4+ε
(2 + δ )2
whenever
ⱍ
4
ⱍ
0 < x 2 < .
(2 − δ )2
ⱍ
ⱍ
ⱍ
ⱍ ⱍ
ⱍⱍ
ⱍ
To find an appropriate , begin by writing x2 4 x 2 x 2 . For all x in the
interval 共1, 3兲, x 2 < 5 and thus x 2 < 5. So, letting be the minimum of 兾5
and 1, it follows that, whenever 0 < x 2 < , you have
4−ε
ⱍ
2+δ
2
2−δ
The limit of f 共x兲 as x approaches 2 is 4.
Figure 1.15
ⱍ
ⱍx2 4ⱍ ⱍx 2ⱍⱍx 2ⱍ < 冢5冣共5兲 .
As you can see in Figure 1.15, for x-values within of 2 共x 2兲, the values of f 共x兲 are
within of 4.
Throughout this chapter, you will use the - definition of limit primarily to prove
theorems about limits and to establish the existence or nonexistence of particular types
of limits. For finding limits, you will learn techniques that are easier to use than the -
definition of limit.
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1.2
1.2 Exercises
See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.
Estimating a Limit Numerically In Exercises 1–6,
complete the table and use the result to estimate the limit. Use
a graphing utility to graph the function to confirm your result.
1. lim
x→4
3.9
3.99
f 共x兲
4
4.001
4.01
x6
x→6
x→0
3.999
冪10 x 4
11. lim
13. lim
x4
x 2 3x 4
x
x→3
12. lim
关x兾共x 1兲兴 共2兾3兲
x2
14. lim
tan x
tan 2x
x→2
sin 2x
x
x→0
Finding a Limit Graphically In Exercises 15–22, use the
graph to find the limit (if it exists). If the limit does not exist,
explain why.
4.1
?
15. lim 共4 x兲
16. lim sec x
x→3
2. lim
55
Finding Limits Graphically and Numerically
x→0
y
y
x3
x2 9
4
2
3
2.9
x
2.99
2.999
f 共x兲
3
3.001
3.01
3.1
2
?
1
x
3. lim
1
冪x 1 1
x
x→0
2
3
4
17. lim f 共x兲
0.1
0.01
0.001
f 共x兲
0
0.001
0.01
0.1
x→1
冦
4 x,
f 共x兲 0,
?
x2
x2
f 共x兲 冦x2, 3,
y
4
关1兾共x 1兲兴 共1兾4兲
x→3
x3
x1
x1
2
y
4. lim
x
π
2
18. lim f 共x兲
x→2
x
−π
2
6
3
2
2.9
x
2.99
f 共x兲
2.999
3
3.001
3.01
3.1
2
1
x
x
?
1
sin x
x→0
x
19. lim
5. lim
2
3
ⱍx 2ⱍ
20. lim
x2
x→2
−2
4
x→5
y
0.1
x
0.01
0.001
f 共x兲
0
0.001
0.01
0.1
?
2
4
2
x5
y
3
2
1
6
4
2
x
3 4 5
cos x 1
6. lim
x→0
x
0.1
x
f 共x兲
−2
−3
0.01
0.001
0
0.001
0.01
0.1
?
21. lim cos
x→0
create a table of values for the function and use the result to
estimate the limit. Use a graphing utility to graph the function
to confirm your result.
x→1
x2
x2 x 6
x4 1
x→1 x6 1
9. lim
1
x
22. lim tan x
x→ 兾2
y
y
1
Estimating a Limit Numerically In Exercises 7–14,
7. lim
x
6 8 10
−2
−4
−6
8. lim
x→4
x4
x2 9x 20
2
1
x
−1
1
−π
2
π
2
π
3π
2
x
−1
x3 27
x→3 x 3
10. lim
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
56
Chapter 1
Limits and Their Properties
Graphical Reasoning In Exercises 23 and 24, use the
graph of the function f to decide whether the value of the given
quantity exists. If it does, find it. If not, explain why.
23. (a) f 共1兲
30. Finding a ␦ for a Given ␧
f 共x兲 y
(b) lim f 共x兲
(c) f 共4兲
1
x1
ⱍ
ⱍ
x→4
f
2.0
x
−1
1.01
1.00
0.99
1.5
1 2 3 4 5 6
24. (a) f 共2兲
1.0
201 2 199
101
99
0.5
y
x
(b) lim f 共x兲
4
3
2
x→2
(c) f 共0兲
1
x→0
−2 −1
(e) f 共2兲
x
1 2 3 4 5
−2
(f) lim f 共x兲
2
31. Finding a ␦ for a Given ␧
(d) lim f 共x兲
f 共x兲 2 ⱍ
y
2
Limits of a Piecewise Function In Exercises 25 and 26,
f
1
2
x
x 2
2 < x < 4
x 4
1
lim f 共x兲 4
f 共2兲 0
f 共2兲 6
lim f 共x兲 0
lim f 共x兲 does not exist.
x→2
x→2
29. Finding a ␦ for a Given ␧ The graph of f 共x兲 x 1 is
shown in the figure. Find such that if 0 < x 2 < , then
f 共x兲 3 < 0.4.
ⱍ
ⱍ
ⱍ
ⱍ
5
f
2
x
2.0 2.5
1.6 2.4
3.0
ⱍ
冢
ⱍ
x
3
冣
34. lim 6 35. lim 共x 2 3兲
36. lim 共x 2 6兲
x→2
x→6
x→4
Using the ␧ -␦ Definition of Limit In Exercises 37–48,
find the limit L. Then use the ␧ -␦ definition to prove that the
limit is L.
37. lim 共x 2兲
38. lim 共4x 5兲
x→2
共34 x 1兲
39. lim
40. lim
41. lim 3
42. lim 共1兲
3
43. lim 冪
x
44. lim 冪x
x→0
1.5
ⱍ
33. lim 共3x 2兲
x→6
3
ⱍ
ⱍ
共12 x 1兲
x→4
1.0
The graph of
Finding a ␦ for a Given ␧ In Exercises 33–36, find the
limit L. Then find ␦ > 0 such that f 冇x冈 ⴚ L < 0.01
whenever 0 < x ⴚ c < ␦.
x→4
y
4
4
Figure for 32
ⱍ
x→2
x→2
lim f 共x兲 3
3
is shown in the figure. Find such that if 0 < x 2 < ,
then f 共x兲 3 < 0.2.
ⱍ
28. f 共2兲 0
x→0
2
f 共x兲 x 2 1
of a function f that satisfies the given values. (There are many
correct answers.)
0.5
1
Figure for 31
Sketching a Graph In Exercises 27 and 28, sketch a graph
27. f 共0兲 is undefined.
x
2
32. Finding a ␦ for a Given ␧
x < 0
0 x
x > 3.2
3
2.8
1
x→c
sin x,
26. f 共x兲 1 cos x,
cos x,
f
4
3
sketch the graph of f. Then identify the values of c for which
lim f 冇x冈 exists.
冦
冦
ⱍ
y
1.1
1
0.9
x→4
2.6
The graph of
ⱍ
(h) lim f 共x兲
x2,
25. f 共x兲 8 2x,
4,
4
1
x
ⱍ
(g) f 共4兲
3.4
3
is shown in the figure. Find such that if 0 < x 1 < ,
then f 共x兲 1 < 0.1.
x→2
ⱍ
ⱍ
y
3
2
1
(d) lim f 共x兲
ⱍ
is shown in the figure. Find such that if 0 < x 2 < ,
then f 共x兲 1 < 0.01.
6
5
x→1
The graph of
x→3
x→2
ⱍ
ⱍ
x→4
ⱍ
ⱍ
45. lim x 5
46. lim x 3
47. lim 共x 1兲
48. lim 共x 4x兲
x→5
2
x→1
x→3
2
x→4
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1.2
49. Finding a Limit
approaches ?
What is the limit of f 共x兲 4 as x
50. Finding a Limit
approaches ?
What is the limit of g共x兲 x as x
Writing In Exercises 51–54, use a graphing utility to graph
the function and estimate the limit (if it exists). What is the
domain of the function? Can you detect a possible error in
determining the domain of a function solely by analyzing the
graph generated by a graphing utility? Write a short paragraph
about the importance of examining a function analytically as
well as graphically.
51. f 共x兲 冪x 5 3
52. f 共x兲 x4
lim f 共x)
x3
x 2 4x 3
57
WRITING ABOUT CONCEPTS
57. Describing Notation
meaning of the notation
Write a brief description of the
lim f 共x兲 25.
x→8
58. Using the Definition of Limit The definition of
limit on page 52 requires that f is a function defined on an
open interval containing c, except possibly at c. Why is this
requirement necessary?
59. Limits That Fail to Exist Identify three types of
behavior associated with the nonexistence of a limit.
Illustrate each type with a graph of a function.
60. Comparing Functions and Limits
lim f 共x兲
x→4
Finding Limits Graphically and Numerically
(a) If f 共2兲 4, can you conclude anything about the limit
of f 共x兲 as x approaches 2? Explain your reasoning.
x→3
x9
53. f 共x兲 冪x 3
(b) If the limit of f 共x兲 as x approaches 2 is 4, can you
conclude anything about f 共2兲? Explain your reasoning.
lim f 共x兲
x→9
54. f 共x兲 x3
x2 9
61. Jewelry A jeweler resizes a ring so that its inner circumference is 6 centimeters.
lim f 共x兲
x→3
(a) What is the radius of the ring?
55. Modeling Data For a long distance phone call, a hotel
charges $9.99 for the first minute and $0.79 for each additional
minute or fraction thereof. A formula for the cost is given by
C共t兲 9.99 0.79 冀 共t 1兲冁
(c) Use the - definition of limit to describe this situation.
Identify and .
where t is the time in minutes.
共Note: 冀x冁 greatest integer n such that n
冀3.2冁 3 and 冀1.6冁 2.兲
x. For example,
(a) Use a graphing utility to graph the cost function for
0 < t 6.
(b) Use the graph to complete the table and observe the
behavior of the function as t approaches 3.5. Use the graph
and the table to find lim C 共t兲.
t→3.5
3
t
3.3
3.4
3.5
3.6
3.7
4
?
C
(c) Use the graph to complete the table and observe the
behavior of the function as t approaches 3.
t
2
2.5
2.9
C
3
3.1
3.5
4
?
Does the limit of C共t兲 as t approaches 3 exist? Explain.
56. Repeat Exercise 55 for
C共t兲 5.79 0.99冀 共t 1兲冁.
(b) The inner circumference of the ring varies between
5.5 centimeters and 6.5 centimeters. How does the radius
vary?
62. Sports
A sporting goods manufacturer designs a golf ball having a
volume of 2.48 cubic inches.
(a) What is the radius
of the golf ball?
(b) The volume of the
golf ball varies
between 2.45 cubic
inches and 2.51 cubic
inches. How does the
radius vary?
(c) Use the - definition of limit to describe this situation.
Identify and .
63. Estimating a Limit Consider the function
f 共x兲 共1 x兲1兾x.
Estimate
lim 共1 x兲1兾x
x→0
by evaluating f at x-values near 0. Sketch the graph of f.
The symbol
indicates an exercise in which you are instructed to use graphing
technology or a symbolic computer algebra system. The solutions of other exercises may
also be facilitated by the use of appropriate technology.
Tony Bowler/Shutterstock.com
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
58
Chapter 1
Limits and Their Properties
64. Estimating a Limit Consider the function
f 共x兲 73. Evaluating Limits Use a graphing utility to evaluate
ⱍx 1ⱍ ⱍx 1ⱍ.
lim
x
x→0
Estimate
lim
for several values of n. What do you notice?
ⱍx 1ⱍ ⱍx 1ⱍ
74. Evaluating Limits Use a graphing utility to evaluate
x
x→0
lim
by evaluating f at x-values near 0. Sketch the graph of f.
x→0
65. Graphical Analysis The statement
x2 4
4
x→2 x 2
means that for each > 0 there corresponds a > 0 such that
if 0 < x 2 < , then
ⱍ
ⱍ
x2 4
4 < .
x2
If
ⱍ
ⱍ
tan nx
x
for several values of n. What do you notice?
lim
ⱍ
sin nx
x
75. Proof Prove that if the limit of f 共x兲 as x approaches c exists,
then the limit must be unique. 关Hint: Let lim f 共x兲 L1 and
x→c
lim f 共x兲 L 2 and prove that L1 L2.兴
x→c
76. Proof Consider the line f 共x兲 mx b, where m 0. Use
the - definition of limit to prove that lim f 共x兲 mc b.
x→c
77. Proof
Prove that
lim f 共x兲 L
0.001, then
x→c
ⱍ
is equivalent to
x2 4
4 < 0.001.
x2
lim 关 f 共x兲 L兴 0.
Use a graphing utility to graph each side of this inequality. Use
the zoom feature to find an interval 共2 , 2 兲 such that
the graph of the left side is below the graph of the right side of
the inequality.
x→c
78. Proof
(a) Given that
lim 共3x 1兲共3x 1兲x2 0.01 0.01
x→0
HOW DO YOU SEE IT? Use the graph of f to
66.
identify the values of c for which lim f 共x兲 exists.
x→c
(a)
(b)
y
(b) Given that lim g 共x兲 L, where L > 0, prove that there
y
x→c
6
6
exists an open interval 共a, b兲 containing c such that g共x兲 > 0
for all x c in 共a, b兲.
4
4
2
−4
x
2
4
6
x
−2
−2
2
prove that there exists an open interval 共a, b兲 containing 0
such that 共3x 1兲共3x 1兲x2 0.01 > 0 for all x 0 in
共a, b兲.
4
True or False? In Exercises 67–70, determine whether the
PUTNAM EXAM CHALLENGE
79. Inscribe a rectangle of base b and height h in a circle of
radius one, and inscribe an isosceles triangle in a region of
the circle cut off by one base of the rectangle (with that
side as the base of the triangle). For what value of h do the
rectangle and triangle have the same area?
statement is true or false. If it is false, explain why or give an
example that shows it is false.
67. If f is undefined at x c, then the limit of f 共x兲 as x approaches
c does not exist.
68. If the limit of f 共x兲 as x approaches c is 0, then there must exist
a number k such that f 共k兲 < 0.001.
h
b
69. If f 共c兲 L, then lim f 共x兲 L.
x→c
70. If lim f 共x兲 L, then f 共c兲 L.
x→c
Determining a Limit In Exercises 71 and 72, consider the
function f 冇x冈 ⴝ 冪x.
71. Is lim 冪x 0.5 a true statement? Explain.
x→0.25
72. Is lim 冪x 0 a true statement? Explain.
80. A right circular cone has base of radius 1 and height 3. A
cube is inscribed in the cone so that one face of the cube is
contained in the base of the cone. What is the side-length
of the cube?
These problems were composed by the Committee on the Putnam Prize Competition.
© The Mathematical Association of America. All rights reserved.
x→0
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
A12
Answers to Odd-Numbered Exercises
13. (a) x ⬇ 1.2426, ⫺7.2426
(b) 共x ⫹ 3兲2 ⫹ y2 ⫽ 18
y
3.
4
3
y
2
8
1
x
−4 −3 −2 −1
−1
1
2
3
6
4
−2
2
−3
−8
−4
(a)
(b)
y
4
3
3
2
2
1
1
2
3
15. Proof
y
x
−4 −3 −2 −1
−1
4
1
2
3
4
2
−2
−3
−3
−4
−4
(c)
(d)
y
4
−6
x
1
2
y
4
−4 −3 −2 −1
−1
x
−4 −2
−2
(− 2 , 0)
4
3
3
2
2
( 2 , 0)
x
−2
2
y
4
1
(0, 0)
−1
−2
1
x
x
−4 −3 −2 −1
−1
1
2
3
−4 −3 −2 −1
−1
4
−2
−2
−3
−3
−4
−4
(e)
(f)
y
1
2
3
4
4
3
3
2
1
1
x
−4 −3 −2 −1
−1
1
2
3
x
−4 −3 −2 −1
−1
4
−2
−2
−3
−3
−4
−4
1
2
3
4
5. (a) A共x兲 ⫽ x 关共100 ⫺ x兲兾2兴; Domain: 共0, 100兲
(b) 1600
Dimensions 50 m ⫻ 25 m
yield maximum area of
1250 m2.
0
(page 47)
Section 1.1
y
4
Chapter 1
110
0
(c) 50 m ⫻ 25 m; Area ⫽ 1250 m2
7. T共x兲 ⫽ 关2冪4 ⫹ x2 ⫹ 冪共3 ⫺ x兲2 ⫹ 1兴兾4
9. (a) 5, less (b) 3, greater (c) 4.1, less
(d) 4 ⫹ h (e) 4; Answers will vary.
11. (a) Domain: 共⫺ ⬁, 1兲 傼 共1, ⬁兲; Range: 共⫺ ⬁, 0兲 傼 共0, ⬁兲
x⫺1
(b) f 共 f 共x兲兲 ⫽
x
Domain: 共⫺ ⬁, 0兲 傼 共0, 1兲 傼 共1, ⬁兲
(c) f 共 f 共 f 共x兲兲兲 ⫽ x
Domain: 共⫺ ⬁, 0兲 傼 共0, 1兲 傼 共1, ⬁兲
y
(d)
The graph is not a line
because
there are holes at
2
x ⫽ 0 and x ⫽ 1.
1. Precalculus: 300 ft
3. Calculus: Slope of the tangent line at x ⫽ 2 is 0.16.
5. (a) Precalculus: 10 square units
(b) Calculus: 5 square units
y
7. (a)
(b) 1; 32; 52
10
(c) 2. Use points closer to P.
P
8
6
x
−2
2
4
8
9. Area ⬇ 10.417; Area ⬇ 9.145; Use more rectangles.
Section 1.2
1.
3.9
3.99
3.999
4
f 共x兲
0.2041
0.2004
0.2000
?
x
4.001
4.01
4.1
f 共x兲
0.2000
0.1996
0.1961
x
lim
x→4
3.
(page 55)
x⫺4
1
⬇ 0.2000 Actual limit is .
x2 ⫺ 3x ⫺ 4
5
冢
⫺0.1
⫺0.01
⫺0.001
0
f 共x兲
0.5132
0.5013
0.5001
?
x
0.001
0.01
0.1
f 共x兲
0.4999
0.4988
0.4881
x
冣
1
x
−2
1
−2
2
lim
x→0
冪x ⫹ 1 ⫺ 1
x
⬇ 0.5000
冢Actual limit is 12.冣
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
A13
Answers to Odd-Numbered Exercises
5.
x
f 共x兲
⫺0.1
⫺0.01
⫺0.001
0
0.9983
0.99998
1.0000
?
25.
6
5
5
4
4
3
x
0.001
0.01
0.1
f 共x兲
1.0000
0.99998
0.9983
lim
x→0
7.
9.
1
2
3
4
x
5
−2 −1
−1
0.2564
0.2506
0.2501
?
x
1.001
1.01
1.1
f 共x兲
0.2499
0.2494
0.2439
29.
33.
35.
41.
51.
冢
0.9
0.99
0.999
1
0.7340
0.6733
0.6673
?
1
31. ␦ ⫽ 11
␦ ⫽ 0.4
⬇ 0.091
L ⫽ 8. Let ␦ ⫽ 0.01兾3 ⬇ 0.0033.
37. 6
L ⫽ 1. Let ␦ ⫽ 0.01兾5 ⫽ 0.002.
3
43. 0
45. 10
47. 2
49. 4
0.5
53. 10
1.1
f 共x兲
0.6660
0.6600
0.6015
lim
x→1
11.
x4 ⫺ 1
2
⬇ 0.6666 Actual limit is .
x6 ⫺ 1
3
冢
⫺6.1
x
⫺6.01
−6
f 共x兲
⫺0.1248 ⫺0.1250 ⫺0.1250
x
⫺5.999
0
f 共x兲
⫺0.1250 ⫺0.1250 ⫺0.1252
13.
15.
19.
21.
23.
冪10 ⫺ x ⫺ 4
⬇ ⫺0.1250
lim f 共x兲 ⫽ 6
x
⫺0.1
⫺0.01
⫺0.001
0
f 共x兲
1.9867
1.9999
2.0000
?
x
0.001
0.01
0.1
f 共x兲
2.0000
1.9999
1.9867
Domain: 关0, 9兲 傼 共9, ⬁兲
The graph has a hole
at x ⫽ 9.
⫺6
0
?
sin 2x
⬇ 2.0000 (Actual limit is 2.)
lim
x→0
x
1
17. 2
Limit does not exist. The function approaches 1 from the right
side of 2, but it approaches ⫺1 from the left side of 2.
Limit does not exist. The function oscillates between 1 and ⫺1
as x approaches 0.
(a) 2
(b) Limit does not exist. The function approaches 1 from the
right side of 1, but it approaches 3.5 from the left side of 1.
(c) Value does not exist. The function is undefined at x ⫽ 4.
(d) 2
6
8
冢Actual limit is ⫺ 81.冣
x⫹6
10
x→9
Domain: 关⫺5, 4兲 傼 共4, ⬁兲
The graph has a hole
at x ⫽ 4.
55. (a) 16
⫺5.9
x→⫺6
39. ⫺3
0
lim f 共x兲 ⫽ 16
(b)
lim
5
6
− 0.1667
冣
⫺6.001
⫺5.99
4
冣
x→4
1.01
3
on the graph except where
c ⫽ 4.
x⫺2
1
⬇ 0.2500 Actual limit is .
x2 ⫹ x ⫺ 6
4
1.001
2
lim f 共x兲 exists for all points
f 共x兲
x
1
x→c
1
f 共x兲
1
−2
0.999
x
1
x
0.99
lim
2
−2 −1
−1
0.9
x→1
f
f
2
sin x
⬇ 1.0000 (Actual limit is 1.)
x
x
y
27.
y
6
t
3
3.3
3.4
3.5
C
11.57
12.36
12.36
12.36
t
3.6
3.7
4
C
12.36
12.36
12.36
lim C共t兲 ⫽ 12.36
t→3.5
(c)
t
2
2.5
2.9
3
C
10.78
11.57
11.57
11.57
t
3.1
3.5
4
C
12.36
12.36
12.36
The limit does not exist because the limits from the right
and left are not equal.
57. Answers will vary. Sample answer: As x approaches 8 from
either side, f 共x兲 becomes arbitrarily close to 25.
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
A14
Answers to Odd-Numbered Exercises
59. (i) The values of f approach
(ii) The values of f increase
different numbers as x
or decrease without
approaches c from different
bound as x
sides of c.
approaches c.
y
6
3
5
2
4
1
3
x
−4 −3 −2 −1
−1
1
2
3
x
−3 −2 −1
−1
−4
−2
2
3
4
5
(iii) The values of f oscillate between two fixed numbers as x
approaches c.
y
5.
17.
25.
33.
39.
41.
4
−4
3
x→0
x2 ⫺ 1
43. f 共x兲 ⫽
and g共x兲 ⫽ x ⫺ 1 agree except at x ⫽ ⫺1.
x⫹1
lim f 共x兲 ⫽ lim g共x兲 ⫽ ⫺2
x
2
4
x→⫺1
−3
−4
3
⬇ 0.9549 cm
␲
5.5
6.5
(b)
, or approximately 0.8754 < r < 1.0345
ⱕr ⱕ
2␲
2␲
(c) lim 2␲ r ⫽ 6; ␧ ⫽ 0.5; ␦ ⬇ 0.0796
61. (a) r ⫽
r → 3兾␲
63.
␲
(a) 0 (b) ⫺5
(a) 0 (b) About 0.52 or ␲兾6
8
7. ⫺1
9. 0
11. 7
13. 2
15. 1
19. 1兾5
21. 7
23. (a) 4 (b) 64 (c) 64
1兾2
(a) 3 (b) 2 (c) 2
27. 1
29. 1兾2
31. 1
35. ⫺1
37. (a) 10 (b) 5 (c) 6 (d) 3兾2
1兾2
(a) 64 (b) 2 (c) 12 (d) 8
x2 ⫹ 3x
and g共x兲 ⫽ x ⫹ 3 agree except at x ⫽ 0.
f 共x兲 ⫽
x
lim f 共x兲 ⫽ lim g共x兲 ⫽ 3
x→0
3
−4 −3 −2
−␲
8
−6
1
−3
4
3.
−4
2
4
6
1.
y
4
(page 67)
Section 1.3
x
⫺0.001
⫺0.0001
⫺0.00001
f 共x兲
2.7196
2.7184
2.7183
x
0.00001
0.0001
0.001
f 共x兲
2.7183
2.7181
2.7169
x→⫺1
x3 ⫺ 8
45. f 共x兲 ⫽
and g共x兲 ⫽ x 2 ⫹ 2x ⫹ 4 agree except at x ⫽ 2.
x⫺2
lim f 共x兲 ⫽ lim g共x兲 ⫽ 12
x→2
x→2
47. ⫺1
49. 1兾8
51. 5兾6
53. 1兾6
55. 冪5兾10
57. ⫺1兾9
59. 2
61. 2x ⫺ 2
63. 1兾5
65. 0
67. 0
69. 0
71. 1
73. 3兾2
2
75.
The graph has a hole at x ⫽ 0.
−3
3
−2
lim f 共x兲 ⬇ 2.7183
x→0
Answers will vary. Sample answer:
x
⫺0.1
⫺0.01
⫺0.001
0.001
0.01
0.1
f 共x兲
0.358
0.354
0.354
0.354
0.353
0.349
y
7
lim
冪x ⫹ 2 ⫺ 冪2
x
x→0
77.
3
⬇ 0.354; Actual limit is
3
(0, 2.7183)
The graph has a hole at x ⫽ 0.
2
1
−5
x
−3 −2 −1
−1
65.
1
2
3
4
1
5
0.002
(1.999, 0.001)
(2.001, 0.001)
1.998
冪2
1
⫽
.
4
2冪2
2.002
67. False. The existence or
nonexistence of f 共x兲 at
x ⫽ c has no bearing on
the existence of the
limit of f 共x兲 as x → c.
0
␦ ⫽ 0.001, 共1.999, 2.001兲
69. False. See Exercise 17.
71. Yes. As x approaches 0.25 from either side, 冪x becomes
arbitrarily close to 0.5.
sin nx
73. lim
75–77. Proofs
⫽n
x→0
x
79. Putnam Problem B1, 1986
−2
Answers will vary. Sample answer:
x
f 共x兲
x
f 共x兲
lim
x→0
⫺0.1
⫺0.01
⫺0.001
⫺0.263
⫺0.251
⫺0.250
0.001
0.01
0.1
⫺0.250
⫺0.249
⫺0.238
1
关1兾共2 ⫹ x兲兴 ⫺ 共1兾2兲
⬇ ⫺0.250; Actual limit is ⫺ .
x
4
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