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Q. $\displaystyle\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}\right)$

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos 2(x-2)}}{x-2}\right)$
$=\displaystyle\lim _{x \rightarrow 2} \frac{\sqrt{2}|\sin (x-2)|}{x-2}$
which doesn't exist as $LHL =-\sqrt{2}$ whereas
$RHL =\sqrt{2}$