Thomas Calculus 13th [Solutions]

Section 2.6 Limits Involving Infinity; Asymptotes of Graphs 99
83.
2
lim 2x 4x 3x
2
x
x
2
x
2
x
3x
2
x
2
2x
4
3 2
2 x
x
2
x
2 2 4 3 2
lim 2x 4x 3x
2
x 2x 4x 3x
2
lim 3x
2
x
2
2x 4x 3x
2
lim
x
3
2 4
2
x
3 2
x
x
2
lim
x
3 0 3
2 2 4
2 2
(4 x ) (4x 3x
2)
lim
2
x 2x 4x 3x
2
lim
x
3x
2
x
2x
4
3 2
x x
x
2
84.
lim
2
9x x 3x
x
2
2
2 2
2 9x x 3x
(9 ) (9 )
lim 9x x 3x
lim x x x
2
x 9x x 3x
x 9x x 3x
x
x
lim
lim 1 1 1
9x 2
x 3x
x 9 3 3 3 6
1
x
x
2
x
2 x
x
lim x
x
2
9x x 3x
85.
lim
2
x 3x 2
x 2x
x
2 2
2 2 x 3x x 2x
lim x 3x x 2x
x
2 2
x 3x x 2x
lim 5x
lim 5 5 5
x
2 2
3 2
x 3x x 2x
x 1 1
1 1 2
x
x
2 2
( 3 ) ( 2 )
lim x x x x
2 2
x x 3x x 2x
86.
lim
x
2 2
x x x x
2 2
2 2 x x x x
lim x x x x
x
2 2
x x x x
lim 2 2
1
1
1
1 1 1
1
x
x
x
2 2
( ) ( )
lim x x x x
2 2
x x x x x
lim 2x
x
2
x x
2
x x
87. For any 0, take N 1. Then for all x N we have that f ( x) k k k 0 .
88. For any 0, take N 1. Then for all y N we have that f ( x) k k k 0 .
89. For every real number B 0, we must find a 0 such that for all x, 0 x 0 1 B.
2
x
Now, 1 B 2
0 1 B 0 x 1 1 .
2 2
x
x
B
x Choose 1 , then 0 x x 1
B
B B
1 B so that lim 1 .
2
2
x
x 0 x
90. For every real number B 0, we must find a 0 such that for all x, 0 x 0 1
x
B . Now,
1 B 0 x 1 . Choose 1 .
x
B
B Then 0 0 1 1
B x
lim 1
x 0 x
.
91. For every real number B 0, we must find a 0 such that for all x, 0 x 3 2 B . Now,
2
( x 3)
2
2 B 0 2 ( x 3) 1
2
B 0
( 3) 2 0 3 2 .
2
2
( x 3)
( x 3)
2 B
x B
X B
Choose 2 B
, then
0 x 3 2 B 0 so that lim 2 .
2
2
( x 3)
x 3 ( x 3)
92. For every real number B 0, we must find a 0 such that for all x, 0 x ( 5) 1 B.
2
( x 5)
Now, 1 2
B 0 ( x 5) 1 x 5 1 . Choose 1 . Then 0
2
( x 5)
B
x ( 5)
B
B x 5 1 1 B so that lim 1 .
2
B ( x 5)
x 5 ( x 5)
2
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