_ x 0 , ln ,( x) x^2 is equal to - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} \,\frac{{{\rm{ln}}\,(\cos x)}}{{{x^2}}}$ is equal to

A
$0$
B
$1$
C
$\frac{1}{2}$
✓
$ - \frac{1}{2}$

✓

Answer

Correct option: D.

$ - \frac{1}{2}$

d
(d) Applying $L-$ Hospital’s rule,

$\mathop {\lim }\limits_{x \to 0} \frac{{\ln (\cos x)}}{{{x^2}}}$$ = \mathop {\lim }\limits_{x \to 0} \frac{{ - \tan x}}{{2x}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{ - {{\sec }^2}x}}{2} = \frac{{ - 1}}{2}$.

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