James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

114 Chapter 2 Limits and Derivatives
(b) If the temperature is allowed to vary from 200°C by
up to 61°C, what range of wattage is allowed for the
input power?
(c) In terms of the «, definition of lim x l a f sxd 5 L, what
is x? What is f sxd? What is a? What is L? What value of
« is given? What is the corresponding value of ?
13. (a) Find a number such that if | x 2 2 |
| 4x 2 8 | , «, where « 5 0.1.
(b) Repeat part (a) with « 5 0.01.
, , then
14. Given that lim x l 2 s5x 2 7d 5 3, illustrate Definition 2 by
finding values of that correspond to « 5 0.1, « 5 0.05, and
« 5 0.01.
15–18 Prove the statement using the «, definition of a limit and
illustrate with a diagram like Figure 9.
15. lim s1 1 1
x l 3
3 xd − 2 16. lim s2x 2 5d 5 3
x l 4
17.
lim s1 2 4xd − 13 18. lim s3x 1 5d − 21
xl23 xl22
19–32 Prove the statement using the «, definition of a limit.
2 1 4x
19. lim 5 2 20. lim
x l1 3
s3 2 4 5 xd − 25
x l 10
x 2 2 2x 2 8
9 2 4x 2
21. lim
− 6 22. lim
xl4 x 2 4
x l21.5 3 1 2x − 6
23.
lim x − a 24. lim c − c
x l a x l a
25. lim
x l 0
x 2 − 0
26. lim
x l 0
x 3 − 0
27. lim
x l 0
| x | − 0 28. lim 6 1 x − 0
xl26 1 s8
29. lim
x l 2
sx 2 2 4x 1 5d 5 1
31.
lim
x l22 sx 2 2 1d 5 3
30. lim
x l 2
sx 2 1 2x 2 7d 5 1
32. lim
x l 2
x 3 5 8
33. Verify that another possible choice of for showing that
lim x l3 x 2 5 9 in Example 4 is 5 minh2, «y8j.
34. Verify, by a geometric argument, that the largest possible choice
of for showing that lim x l3 x 2 − 9 is − s9 1 « 2 3.
CAS
35. (a) For the limit lim x l 1 sx 3 1 x 1 1d 5 3, use a graph to
find a value of that corresponds to « 5 0.4.
(b) By using a computer algebra system to solve the cubic
equation x 3 1 x 1 1 5 3 1 «, find the largest possible
value of that works for any given « . 0.
(c) Put « 5 0.4 in your answer to part (b) and compare
with your answer to part (a).
1
36. Prove that lim
x l2 x − 1 2 .
37. Prove that lim
x l a
sx − sa if a . 0.
FHint: Use
| sx 2 sa | −
| x 2 a |
sx 1 sa
.
38. If H is the Heaviside function defined in Example 2.2.6,
prove, using Definition 2, that lim t l 0 Hstd does not exist.
[Hint: Use an indirect proof as follows. Suppose that the
limit is L. Take « 5 1 2 in the definition of a limit and try to
arrive at a contradiction.]
39. If the function f is defined by
f sxd −H 0 1
if x is rational
if x is irrational
prove that lim x l 0 f sxd does not exist.
40. By comparing Definitions 2, 3, and 4, prove Theorem 2.3.1.
41. How close to 23 do we have to take x so that
1
sx 1 3d 4 . 10,000
42. Prove, using Definition 6, that lim
x l23
1
sx 1 3d 4 − `.
43. Prove that lim ln x − 2`.
x l 01 44. Suppose that lim x l a f sxd 5 ` and lim x l a tsxd 5 c, where
c is a real number. Prove each statement.
(a) lim f f sxd 1 tsxdg − `
x l a
(b) lim f f sxdtsxdg 5 ` if c . 0
x l a
(c) lim f f sxdtsxdg 5 2` if c , 0
xl a
F
We noticed in Section 2.3 that the limit of a function as x approaches a can often be
found simply by calculating the value of the function at a. Functions with this property
are called continuous at a. We will see that the mathematical definition of continuity corresponds
closely with the meaning of the word continuity in everyday language. (A continuous
process is one that takes place gradually, without interruption or abrupt change.)
1 Definition A function f is continuous at a number a if
lim f sxd − f sad
xl a
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