Thomas Calculus 13th [Solutions]

100 Chapter 2 Limits and Continuity
93. (a) We say that f ( x ) approaches infinity as x approaches x 0 from the left, and write lim f ( x) ,
x x0
if for every positive number B, there exists a corresponding number 0 such that for all x,
x0 x x0 f ( x) B.
(b) We say that f ( x ) approaches minus infinity as x approaches x 0 from the right, and write lim f ( x) ,
x x0
if for every positive number B (or negative number B) there exists a corresponding number 0 such
that for all x, x 0 x x 0 f ( x) B.
(c) We say that f ( x ) approaches minus infinity as x approaches x 0 from the left, and write lim f ( x ) , if
x x0
for every positive number B (or negative number B) there exists a corresponding number 0 such that
for all x, x 0 x x 0 f ( x) B.
94. For B 0, 1 0 1 .
x
B x B
Choose 1 B . Then 0 x 0 x 1 1
B x
B so that lim 1 .
x
x 0
95. For B 0, 1 B 0 1 B 0 x 1 1 x . Choose 1 .
x x B B
B Then x 0 1
B
1 B so that lim 1 .
x
x
x 0
x
96. For B 0, 1 1
x 2 B x 2
B ( x 2) 1 x 2 1 x 2 1 . Choose 1 .
B B B
B
Then 2 x 2 x 2 0 1 2 0 1
B
x x 2
B 0 so that lim 1
x 2
.
x 2
97. For B 0, 1 1
x 2
B 0 x 2 B
. Choose 1 B . Then 2 x 2 0 x 2 0 x 2 1
B
1 B 0 so that 1
x 2
lim x 2
.
x 2
98. For B 0 and 0 x 1, 1 B 1 x 2 1 (1 )(1 ) 1 .
2
1 x
B
x x B
Now 1 x 2 1 since x 1. Choose 1
2 B
.
Then 1 x 1 x 1 0 1 x 1 (1 x)(1 x) 1 1 x 1 1 B for 0 x 1 and
2B B 2 B 2
1 x
x near 1 lim 1 . 2
x 1 1 x
99.
2
y x 1 1
x 1 x x 1
100.
y
x
2 1 x 1 2
x 1 x 1
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