lim _ x 2 ( 1 - cos 2 ( x - 2 ) x - 2 )= - Vidyadip

MCQ

$\mathop {{\rm{lim}}}\limits_{x \to 2} \left( {\frac{{\sqrt {1 - {\rm{cos}}\left\{ {2\left( {x - 2} \right)} \right\}} }}{{x - 2}}} \right)=$

A
$\sqrt 2 $
B
-$\;\sqrt 2 $
C
$\frac{1}{{\sqrt 2 }}$
✓
does not exist

✓

Answer

Correct option: D.

does not exist

d
$\mathop {\lim }\limits_{x \to 2} \frac{{\sqrt {1 - \cos \{ 2(x - 2)\} } }}{{x - 2}}$

$ = \mathop {\lim }\limits_{x \to 2} \frac{{\sqrt 2 |\sin (x - 2)|}}{{x - 2}}$

$R.H.L. = \mathop {\lim }\limits_{x \to {2^ - }} \frac{{\sqrt 2 \sin (x - 2)}}{{(x - 2)}} = - \sqrt 2 $

$R.H.L. = \mathop {\lim }\limits_{x \to {2^ + }} \frac{{\sqrt 2 \sin (x - 2)}}{{(x - 2)}} = - \sqrt 2 $

Thus $L . H . L . \neq R . H . L$

Hence, $\mathop {\lim }\limits_{x \to 2} \frac{{\sqrt {1 - \cos \{ 2(x - 2)\} } }}{{x - 2}}$ does not exist.

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