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Q. $\displaystyle\lim_{x \to 0} \frac{ \sqrt{1 -\cos \, 2x}}{\sqrt{2} x}$ is

AIEEEAIEEE 2002Limits and Derivatives

Solution:

$\lim\frac{\sqrt{1-\cos 2x}}{\sqrt{2} x} \Rightarrow \lim \frac{\sqrt{1-\left(1-2 \sin^{2} x\right)}}{\sqrt{2} x} $
$\displaystyle\lim_{x \to0} \frac{\sqrt{2 \sin^{2} x}}{\sqrt{2} x} \Rightarrow \lim_{x \to0} \frac{\left|\sin x\right|}{x} $
The limit of above does not exist as
LHS = -1 $\neq $ RHL = 1