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

104 Chapter 2 Limits and Derivatives
52. Let
tsxd −
x
3
2 2 x 2
x 2 3
if x , 1
if x − 1
if 1 , x < 2
if x . 2
(a) Evaluate each of the following, if it exists.
(i) lim x l1 2 (ii) lim x l 1
(iii) ts1d
(iv) lim x l2 2 (v) lim x l 2 1 (vi) lim x l 2
(b) Sketch the graph of t.
53. (a) If the symbol v b denotes the greatest integer function
defined in Example 10, evaluate
(i) lim v x b (ii) lim v x b (iii) lim v x b
x l22 1 x l22 x l22.4
(b) If n is an integer, evaluate
(i) lim v x b (ii) lim v x b
x ln 2 x l n 1
(c) For what values of a does lim x l a v x b exist?
54. Let f sxd − vcos x b, 2 < x < .
(a) Sketch the graph of f.
(b) Evaluate each limit, if it exists.
(i) lim f sxd (ii) lim f sxd
x l 0 x lsy2d 2
(iii) lim f sxd (iv) lim f sxd
x lsy2d 1 x ly2
(c) For what values of a does lim x l a f sxd exist?
55. If f sxd − v x b 1 v2x b, show that lim x l 2 f sxd exists but is
not equal to f s2d.
56. In the theory of relativity, the Lorentz contraction formula
f sxd 2 8
59. If lim − 10, find lim f sxd.
x l 1 x 2 1
x l 1
60. If lim
x l 0
f sxd
x 2
(a) lim f sxd
x l 0
61. If
− 5, find the following limits.
f sxd −H x 2
prove that lim x l 0 f sxd − 0.
0
(b) lim
x l 0
f sxd
x
if x is rational
if x is irrational
62. Show by means of an example that lim x l a f f sxd 1 tsxdg may
exist even though neither lim x l a f sxd nor lim x l a tsxd exists.
63. Show by means of an example that lim x l a f f sxd tsxdg may
exist even though neither lim x l a f sxd nor lim x l a tsxd exists.
s6 2 x 2 2
64. Evaluate lim
x l 2 s3 2 x 2 1 .
65. Is there a number a such that
3x 2 1 ax 1 a 1 3
lim
x l22 x 2 1 x 2 2
exists? If so, find the value of a and the value of the limit.
66. The figure shows a fixed circle C 1 with equation
sx 2 1d 2 1 y 2 − 1 and a shrinking circle C 2 with radius r
and center the origin. P is the point s0, rd, Q is the upper
point of intersection of the two circles, and R is the point of
intersection of the line PQ and the x-axis. What happens to R
as C 2 shrinks, that is, as r l 0 1 ?
y
L − L 0s1 2 v 2 yc 2
expresses the length L of an object as a function of its velocity
v with respect to an observer, where L 0 is the length of the
object at rest and c is the speed of light. Find lim v lc2L and
interpret the result. Why is a left-hand limit necessary?
57. If p is a polynomial, show that lim xl a psxd − psad.
C
P
0
Q
C¡
R
x
58. If r is a rational function, use Exercise 57 to show that
lim x l a rsxd − rsad for every number a in the domain of r.
The intuitive definition of a limit given in Section 2.2 is inadequate for some purposes
because such phrases as “x is close to 2” and “ f sxd gets closer and closer to L” are vague.
In order to be able to prove conclusively that
lim
x l 0Sx 3 1
we must make the definition of a limit precise.
cos 5x
sin x
− 0.0001 or lim − 1
10,000D x l 0 x
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