Thomas Calculus 13th [Solutions]

Chapter 4 Practice Exercises 327
74. The limit leads to the indeterminate form 0 0 : 3 1 (ln 3)3
lim lim ln 3
1
75. The limit leads to the indeterminate form 0 0 : sin x
sin x
2 1 2 (ln 2)( cos x)
lim lim ln 2
x
x
x e 1 x
e
76. The limit leads to the indeterminate form 0 0 : sin x
sin x
2 1 2 (ln 2)( cos x)
lim lim
x
x
x e 1 x
e
ln 2
77. The limit leads to the indeterminate form 0 0 : lim 5 5cos x lim 5sin x lim 5cos x
x x x
x e x 1 x e 1 x e
5
78. The limit leads to the indeterminate form 0 0 : lim 4 4e
lim 4e
4
x x x
x xe x e xe
x
x
79. The limit leads to the indeterminate form 0 2
0 : t ln(1 2 t)
1
1 2t
lim
lim
2
t
2t
t
t
80. The limit leads to the indeterminate form 0 0 :
2 2
sin ( x) 2 (sin x)(cos x) sin(2 x) 2 cos(2 x) 2
lim lim lim lim 2
x 4 x 4 x 4 x 4
x 4 e 3 x x 4 e 1 x 4 e 1 x 4 e
t t t
81. The limit leads to the indeterminate form 0 0 : lim e 1 lim e 1 lim e 1
t t t
1
t t t
82. The limit leads to the indeterminate form :
1/ y
ln y y y
lim e ln y lim lim lim 0
y
1
y
1
( y
2
)
y
1
y y e y e y e
1
83.
lim 1 b
kx
lim 1
x
x
x
1
x/
b
x/
b
bk
bk
e
84. lim 1 2 7 1 0 0 1
x 2
x
x
85. (a) Maximize
1/2 1/2
f ( x) x 36 x x (36 x ) where 0 x 36
f ( ) 1 1/2 1 1/2 36
2 x x
(36 ) ( 1)
2 2 x 36 x
derivative fails to exist at 0 and 36; f (0) 6, and
f (36) 6 the numbers are 0 and 36
1/2 1/2
1/2
(b) Maximize g( x) x 36 x x (36 x ) where 0 x 36 g ( x)
1 x 1 1/2
(36 ) ( 1)
2 2 36 x x
2 x 36 x
critical points at 0, 18 and 36; g(0) 6, g(18)
2 18 6 2 and g (36) 6 the numbers
are 18 and 18
1/2 3/2
1/2 1/2
20x x where 0 x 20 f ( x)
x 3 x 20 3x
2 2 x
0
x 0 and x 20 are critical points; f (0) f (20) 0 and f 20 20 40 20
20 20
3
3 3 3 3 3
the numbers
and 40
3 .
86. (a) Maximize f ( x) x(20 x)
are 20
3
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