The _ x 0 ( x)^ x is - Vidyadip

MCQ

The $\mathop {\lim }\limits_{x \to 0} {(\cos x)^{\cot x}}$ is

A
$-1$
B
$0$
✓
$1$
D
None of these

✓

Answer

Correct option: C.

$1$

c
(c) $y = \mathop {\lim }\limits_{x \to 0} {(\cos x)^{\cot x}}$

Taking $log$ on both sides,

==> $\log y = \mathop {\lim }\limits_{x \to 0} \,\,\cot x\log \cos x$

==> $\log y = \mathop {\lim }\limits_{x \to 0} \frac{{\log \cos x}}{{\tan x}}$,$\left( {\frac{0}{0} \,\, {\rm{form}}} \right)$

Applying $L-$ Hospital’s rule,

==> $\log y = \mathop {\lim }\limits_{x \to 0} \frac{{ - \tan x}}{{{{\sec }^2}x}}$= 0

==> $y = {e^0}$ ==> $y = 1$.

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