_ x 0 ( 1 + x 1 + x )^ cosec x is equal to - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} {\left( {\frac{{1 + \tan x}}{{1 + \sin x}}} \right)^{{\rm{cosec }}x}}$ is equal to

A
$e$
B
$\frac{1}{e}$
✓
$1$
D
None of these

✓

Answer

Correct option: C.

$1$

c
(c) Given limit $ = \mathop {\lim }\limits_{x \to 0} [{(1 + \tan x)^{\cos ec\,x}} \times 1/{(1 + \sin x)^{\cos ec\,x}}]$

$ = \mathop {\lim }\limits_{x \to 0} \,{[{\{ 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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