Thomas Calculus 13th [Solutions]

210 Chapter 3 Derivatives
124. rabbits/day and foxes/day
125. lim sin x lim sin x 1
2
x 0 2x
x x 0
x (2x
1)
(1) 1 1
1
126. lim 3 x tan 7 x
x 0
2x
lim 3 x sin 7 x
x 0
2x 2x cos 7x
3 lim 1 sin 7x
1
2
x 0
cos7x
7x
2
7
3 1 1 7 2
2 2
127. lim sin r
r 0
tan 2r
lim sin r 2r
1
r 0
r tan 2r
2
1
2
(1) lim cos 2r
r
r 0
sin 2
2r
1 (1) 1 1
2 1 2
128.
sin(sin )
lim lim sin(sin ) sin
0 0
sin
lim sin(sin ) sin
0
sin
lim x 1
x 0
x
lim
0
sin (sin )
. Let x sin . Then x 0 as 0
sin
129.
lim
2
2
4 tan tan 1
2
tan 5
lim
4
1 1
tan
tan
2
1
5
2 tan 2
(4 0 0)
(1 0)
4
130.
lim
0
1 2cot
2
5cot 7cot 8
2
lim
0 5
1
cot
2
2
7 8
cot
cot
2
(0 2) 2
(5 0 0) 5
131. lim x sin x
x 0
2 2cos x
lim x sin x
x 0
2(1 cos x)
lim xsin
x
2
x 0 2 2sin
x
2
lim
x 0
sin
x x
2 2
2 x
2
sin x
x
lim
x 0
sin
x x
2 2
x x
sin
2 2
sin x
x
(1)(1)(1) 1
1 cos
132. lim
2
0
lim
0
2sin
2
2
2
sin
sin
lim 2 2
(1)(1) 1 1
0
2 2
1
2
2 2
133. lim tan x lim 1 sin x 1; let tan x 0 as x 0 lim g( x)
x 0
x
x 0
cos x x
x 0
Therefore, to make g continuous at the origin, define g(0) 1.
tan(tan x)
lim
x 0
tan x
lim tan 1.
0
tan(tan x)
tan(tan )
134. lim f ( x)
lim
sin 1
x 0 x 0
sin(sin x)
lim x x
0
tan x sin(sin x) cos x
1 lim sin x (using the result of # 103); let
x
x 0
sin(sin x)
sin x 0 as x 0 lim sin x lim 1. Therefore, to make f continuous at the origin,
x 0
sin(sin x)
0
sin
define f (0) 1.
135.
2 2
2( x 1) 2( x 1) 2 1 y 2 1 ( 2sin 2 x)
y ln y ln ln(2) ln( x 1) ln(cos 2 x) 0 x
2 2
cos 2x cos 2x y x 1 2 cos 2x
y 2x
2( 1)
tan 2 x y x 2x
tan 2 x
2 2
x 1 cos 2x
x 1
2
136. 10 3 4 10 3 4 1 y
y x ln y ln x [ln(3x 4) ln(2x
4)] 1 3 2
2x 4 2x 4 10 y 10 3x 4 2x
4
y 1 3 1 y 10 3x
4 1 3 1
10 3x 4 x 2 2x 4 10 3x 4 x 2
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