Thomas Calculus 13th [Solutions]

84 Chapter 2 Limits and Continuity
30.
31.
32.
33.
34.
2
lim x x sin x 1 1 sin 1 1
2 lim x x
0
x
0
2 2 2 x
0 2 2
(1) 0
x
x
(1 cos )(1 cos )
2 2
lim 1 cos lim lim 1 cos lim sin
0
sin 2
0
(2sin cos )(1 cos )
0
(2sin cos )(1 cos )
0
(2sin cos )(1 cos )
lim sin 0
0
(2cos )(1 cos ) (2)(2)
0
x(1 cos x) 1 cos x 1 1 cos x
1
cos x x(1 cos x)
9x 2
9x 9
x 0
x
9
2 2
sin
2
3 sin 3
2
sin 3
2 2
x
x
x x x
9x
2 3x
x 0
3x
lim
lim
x x lim lim lim (0)
0
x 0 sin 3 x 0 sin 3 x 0 x 0 lim
1
lim sin(1 cos t)
sin
0
1 cos
lim 1 since 1 cos t 0 as t
t
0
t
0
lim sin(sin h) sin
0
sin
lim 1since sin h 0 as h
h
0
h
0
35. lim sin lim sin 2 1 lim sin 2 1 1 1 1
0
sin 2
0
sin 2 2 2
0
sin 2 2 2
36. lim sin 5x sin 5 4 5 5 sin 5 4 5 5
sin 4 lim x x
0 0
sin 4 5 4 4 lim x x
0
5 sin 4 4 1 1
x
x
x
x x
x
x x
4
37.
lim cos 0 1 0
0
38. lim sin cot 2 lim sin cos 2 lim sin cos 2 lim cos 2 1
0 0
sin 2
0
2sin cos
0
2cos 2
39. lim tan3x lim sin 3x lim sin 3x 8x
3
x 0
sin8x x 0
cos3x sin8x x 0
cos3x sin 8x 3x
8
3 lim 1 sin 3x
8x
3 1 1 1 3
8
x 0
cos3x 3x sin 8x
8 8
40.
41.
42.
sin 3y cot 5y sin 3ysin 4y cos5 y sin 3y sin 4 y cos5 y 3 4 5y
lim lim lim
y 0
y cot 4y y 0
y cos 4 ysin 5y y 0
y cos 4 y sin 5y 3 4 5y
lim sin 3y sin 4 y 5y cos5y
3 4 12 12
0
3 4 sin 5 cos 4 5 1 1 1 1
y
y y y y
5 5
sin
cos
2 2 cos3
2
sin 3
lim tan lim lim sin sin 3 lim sin sin 3 3 3
0 cot 3 0 0 cos cos3 0
3 cos cos3 (1)(1) 11
3
cos 4
2
2
2 2
cot 4 sin 4
cos 4 sin 2
cos 4 (2sin cos )
lim
2 2 2 cos
2
2
2 2 2 2
2 2
sin
0 sin cos 2 sin 4
sin
2
2
2 2 2 2
lim lim lim lim
0 sin cot 2 0 0 sin cos 2 sin 4 0 sin cos 2 sin 4
cos 4 (4sin cos )
4 cos 4 cos 4 cos 4 cos 1 cos 4 cos
lim lim lim 1 11 1
0 cos 2 sin 4 0
sin 4 cos 2 0
cos 2 1 1
2 2 sin 4 2 2
4
43. Yes. If lim f ( x) L lim f ( x ), then lim f ( x) L . If lim f ( x) lim f ( x ), then lim f ( x ) does not
x a x a
x a x a x a
x a
exist.
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