Question

Evaluate : $\underset{x\to 0}{lim}\left(\frac{1-\cos 2x}{x}\right)$

Answer 1

$\underset{x\to 0}{lim}\left(\frac{1-\cos 2x}{x}\right)$
$=\underset{x\to 0}{lim}\left(\frac{1-\cos 2x}{x}\right)$
$=\underset{x\to 0}{lim}\left(\frac{1-1+2{\sin }^{2}x}{x}\right)$
$=\underset{x\to 0}{lim}\frac{2{\sin }^{2}x}{x}$
$=2.\underset{x\to 0}{lim}(\frac{\sin x}{x}.\frac{\sin x}{1})$
$=2.\underset{x\to 0}{lim}\frac{\sin x}{x}.\underset{x\to 0}{lim}\sin x$
$=2\times 1.\sin 0$
$=2\times 0$
$=0$.