Question

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

Answer 1

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

Put $x-\frac{\pi }{6}=t$ as $x\to \frac{\pi }{6},t\to 0$

$={lim}_{t\to 0}\frac{\sin t}{\sqrt{3}-2\cos (t+\frac{\pi }{6})}$

$={lim}_{t\to 0}\frac{\sin t}{\sqrt{3}-\sqrt{3}\cos t+\sin t}$

$={lim}_{t\to 0}\frac{2\sin \frac{t}{2}\cos \frac{t}{2}}{2\sqrt{3}{\sin }^2\left(\frac{t}{2}\right)+2\sin \left(\frac{t}{2}\right)}$

$={lim}_{t\to 0}\frac{\cos \frac{t}{2}}{\sqrt{3}\sin \frac{t}{2}+1}$

$=\frac{\cos 0}{\sqrt{3}\sin 0+1}$

$=\frac{1}{0+1}=1$