Question

$\underset{x\to \frac{\pi }{3}}{lim}\frac{\sqrt{3}\cos x-\sin x}{1-2\cos x}$

Answer 1

$\underset{x\to \frac{\pi }{3}}{lim}\frac{\sqrt{3}\cos x-\sin x}{1-2\cos x}$ $\frac{0}{0}$ form.

Using L'Hospital's rule we get,

$=\underset{x\to \frac{\pi }{3}}{lim}\frac{-\sqrt{3}\sin x-\cos x}{2\sin x}$

$=-\frac{-\frac{3}{2}-\frac{1}{2}}{\sqrt{3}}$

$=-\frac{2}{\sqrt{3}}$