A) \[e\]
B) \[\frac{1}{e}\]
C) 1
D) None of these
Correct Answer: C
Solution :
Given limit \[=\underset{x\to 0}{\mathop{\lim }}\,[{{(1+\tan x)}^{\cos ec\,x}}\times 1/{{(1+\sin x)}^{\cos ec\,x}}]\] \[=\underset{x\to 0}{\mathop{\lim }}\,\,{{[{{\{1+\tan x)}^{\cot \,x}}\}}^{sec\,x}}\times \{1/{{(1+\sin x)}^{\cos ec\,x}}\}]\] \[={{e}^{\sec \,\,0}}.\frac{1}{e}=e\,.\,\frac{1}{e}=1.\]
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