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STD 11 - 12. limits — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 12. limits4 Marks
Question
The $\mathop {\lim }\limits_{x \to 0} {(\cos x)^{\cot x}}$ is
✓

Answer

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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