Thomas Calculus 13th [Solutions]

Section 4.5 Indeterminate Forms and LHôpitals Rule 285
61. The limit leads to the indeterminate form 1 . Let f ( x) x 2
ln f ( x) ln
x 2 x ln
x 2
x
x 1 x 1 x 1
x 2 1 1
3
x 1 x x
x 2 x 1
( x 2)( x 1)
1 1 1 1
x
x
x
2
x
2
ln
2
ln( 2) ln( 1)
f x x x
x x
x 1
x x x x
lim ln ( ) lim ln lim lim lim lim
2
3x
6x
6
x
( x 2)( x 1)
x
2x
1
x
2
lim lim lim 3. Therefore,
2
ln ( ) 3
lim
x
x
f x
lim f ( x) lim e e
x
x 1
x x
x
62. The limit leads to the indeterminate form
0 . Let
2 1/ x
2 1/ x
2
x 1 f x x 1 1 x 1
x 2 x 2 x x 2
f ( x) ln ( ) ln ln
2
ln
x
2 1 2
2 x 1
x 2 ln x 1 ln( x 2)
x
2
1
x 2
2
2
lim ln f ( x) lim
1
ln
x 1
lim lim lim lim
x 4 x 1
x x
x x 2
x
x
x
x
x
1
x ( x 1)( x 2)
lim
2
x 4x 1 lim 2x
4 lim 2
3 2 2
2 2 3 4 1 6x
4
0.
2 1/ x
ln ( )
Therefore, lim
x 1
f x
lim f ( x) lim e
x x x x x x x x
x
x 2
x x
0
e
1
63.
1
x
1 2
x
2
x
3
3 2
2
x x ln x x 3x
2x
2
x 0 x 0 x 0 x 0 x 0
lim ln lim lim lim lim 0
64.
2 1 2
x 2(ln x)
2
x
x x
x
1 1 1 1
x
x
x
2 x
x
2
x x 2 (ln )
2ln 2
x
x 0 x 0 x 0 x 0 x 0 x 0 x 0
lim (ln ) lim lim lim lim lim lim 2 0
x
1 1
2 cot
csc
1
x 0 x 0 x x 0 x
65. lim x tan x lim lim 1
2
2 2
66.
1
x
x x ln x sin x tan x sin xsec x cos x tan x 0
csc x csc x cot x x
1 1
x 0 x 0 x 0 x 0 x 0
lim sin ln lim lim lim lim 0
2
67.
9x
1 9x
1 9
x x 1 x
x 1
x
1
lim lim lim 9 3
x
68.
sin x
x 0 sin x lim
x
x 0
1 1
1
lim 1
69.
sec x
1 cos x
1
x
tan x
x
cos x sin x
x
sin x
lim lim lim 1
2 2 2
70.
cos x
sin x
1
sin x
cot x
x 0
csc x
x 0 x 0
lim lim lim cos x 1
71.
x
x x
2
x
3
1
x x
x 4
x
3
2 3
3 4 1
lim lim 0
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