Thomas Calculus 13th [Solutions]

Section 4.5 Indeterminate Forms and LHôpitals Rule 281
11.
3 2
lim 5x 2x lim 15x 2 lim 30x
30 5
3 2
7 3 21
42 lim
x x x x x
x
x
42 7
12.
2
lim x 8x lim 1 16x
16 2
2
12 5 24 5 lim
x x x x
x
x
24 3
13.
lim 2 cos (2 )
sin lim t t
t
0
t
0
1
0
t
t
2
14. lim sin 5t
lim 5cos5t
5
t 0
2t
t 0
2 2
15.
2
lim 8x
16 16 16
cos 1 lim x
0
x
0
sin x lim 0
cos x 1
16
x x x
16. lim sin x x lim cos x 1 lim sin x cos 1
3 2
0 0 3 0
6 lim x
x x x x x
x
x 0
6 6
17. lim 2 lim 2 2 2
cos(2 ) sin(2 ) 3
sin
2 2
2
18.
19.
lim 3 lim 3 3
sin
cos
3 3 3 3
lim 1 sin lim cos lim sin 1 1
1 cos 2 2sin 2 4cos 2 ( 4)( 1) 4
2 2 2
20. lim x 1 lim 1 1
ln sin( ) 1
x 1
x x
x 1 cos( x)
1
x
21.
2
lim x 2 2 2 2
ln(sec ) lim x lim x
tan lim 2
2 2
x 0
x
x 0 x 0
x
x 0 sec x 1
sec x tan x
sec x
22.
csc x cot x
ln(csc x) csc x
2 2
cot x csc x 1 1
2
2 2
2 2 2
2 x
x
x
2
2 2 2 2
2
lim lim lim lim
x x x x
23.
24.
lim t(1 cos t) (1 cos ) (sin ) sin (sin cos ) cos cos cos sin 1 1 1 0
sin lim t t t
0 0
1 cos lim t t t t
0
sin lim t t t t t
t t t t
0
cost
1
3
t t t t
lim t sin t sin cos cos (cos sin ) 1 (1 0)
0
1 cos lim t t t
0
sin lim t t t t
t t
0
cost
1
2
t t t
25.
x
1 1
x
2
x
cos x
x
sin x 1
lim sec lim 2
x x
lim 1
2 2 2
26.
x
1
x
2
x
cot x
x
csc x
x
lim 2
x tan x lim lim lim sin x 1
2
2 2 2 2
2
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