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Q. $\displaystyle \lim_{x \to2} \left( \frac{\sqrt{ 1- \cos\left\{2\left(x -2\right)\right\}}}{x-2}\right)$

IIT JEEIIT JEE 1998Limits and Derivatives

Solution:

$\displaystyle \lim_{ x\to 2} \frac{\sqrt{1- \cos\left\{2\left(x-2\right)\right\}}}{x-2}$
$ = \displaystyle \lim_{x \to 2} \frac{\sqrt{2} \left|\sin\left(x-2\right)\right|}{x-2}$
$ L.H.L.= \displaystyle \lim _{x \to 2^-} \frac{\sqrt{2} \sin\left(x-2\right)}{\left(x-2\right) } = -1 $
$R.H.L.= \displaystyle \lim_{x \to 2^+} \frac{\sqrt{2} \sin\left(x-2\right)}{\left(x-2\right)} = 1 $
Thus $L.H.L. \ne R.H.L.$
Hence, $\displaystyle \lim _{x \to 2} \frac{\sqrt{1- \cos\left\{2\left(x-2\right)\right\}}}{x-2}$ does not exist.