Question

Evaluate:

${lim}_{x\to \frac{\pi }{3}}\frac{\sqrt{1-\cos 6x}}{\sqrt{2}(\frac{\pi }{3}-x)}$

Answer 1

${lim}_{x\to \frac{\pi }{3}}\frac{\sqrt{1-\cos 6x}}{\sqrt{2}(\frac{\pi }{3}-x)}$

${lim}_{x\to \frac{\pi }{3}}\frac{\sqrt{2{\sin }^{2}3x}}{\sqrt{2}(\frac{\pi }{3}-x)}$ $[\because 1-\cos 2x=2{\sin }^{2}x]$

${lim}_{x\to \frac{\pi }{3}}\frac{\sqrt{2}\sin 3x}{\sqrt{2}(\frac{\pi }{3}-x)}$

${lim}_{x\to \frac{\pi }{3}}\frac{\sin (\pi -3x)}{(\frac{\pi }{3}-x)}$ $[\because \sin (\pi -x)=\sin x]$

${lim}_{x\to \frac{\pi }{3}}\frac{3.\sin (\pi -3x)}{\pi -3x}$

$=3$ $[\because {lim}_{x\to 0}\frac{\sin x}{x}=1]$