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1. lim
tan x � sin x
3
x�0
tan x
x �
1
x
2
� 1
解:原式= lim
� lim �
x�0
x
3
x�0
x
3
2
2.
1
lim(cos
x)
ln(1�2 x
2
)
x�0
(cos x�1)
1
�
1
x
2
�
1
�
ln(1�2 x
2
)
lim
2
1
解:原式= lim
�
(1 � cos x �1)
cos x�1
�
x�0
� �
� e
x�0
�2 x
2
� e
4
3.
arcsin
lim
x
1 � x
2
x��
arctan
x
2x � 1
3x
2
� 4
解:原式= lim
1 � x
2
� lim
x(3x
2
� 4)
�
3
x��
2x � 1
3x
2
� 4
x��
(2x � 1)(1 � x
2
)
2
4.
lim
x�0
e
tan x
� e
sin x
e
sin x
(e
tan x�sin x
� 1)
解:原式=
lim
x�0
= lim
x�0
e
tan x�sin x
� 1
= lim
x�0
tan x �sin x
�1
1
x
3
2
�
sin x
, x � 0
5. f (x) �
�
tan(sin 2 x)
,若 f (x) 在 x � 0 处连续,求 a 。
�
�
解: lim f ( x) � lim
a, x � 0
sin x
�
1
� f (0) � a ,故
a �
1
x�0
x�0
tan(sin 2 x) 2
�
ax
2
sin
1
�
ln(1 � 2 x)
,
2
x � 0
�
6. f
�
x
�
�
�
b,
�
x
1� x
2
sin 2x
x � 0
,若 f (x) 在 x � 0 处连续,求 a, b, c
�
c( )
x
,
x � 0
�
�
1�
x
2
1
ln(1 � 2 x)
解: f (0 � 0) � lim (ax
x�0�
sin �
x