Question

Evaluate:

$\underset{x\to \frac{\pi }{3}}{lim}\frac{\sin \left(\frac{\pi }{3}-x\right)}{2\cos x-1}$

Answer 1

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

Let $x=\frac{\pi }{3}+t$

$\underset{t\to 0}{lim}\frac{\sin (\frac{\pi }{3}-\frac{\pi }{3}-t)}{2\cos (\frac{\pi }{3}+t)-1}$

$=\underset{t\to 0}{lim}\frac{-\sin t}{2\left[\cos \right(t+\frac{\pi }{3}]-1}$

$=\underset{t\to 0}{lim}\frac{-\sin t}{2[\frac{1}{2}\cos t-\frac{\sqrt{3}}{2}\sin t]-1}$

$=\underset{t\to 0}{lim}\frac{-\sin t}{\cos t-\sqrt{3}\sin t-1}$

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

$=\underset{t\to 0}{lim}\frac{2\sin \frac{t}{2}\cos \frac{t}{2}}{\sqrt{3}\times 2\sin \frac{t}{2}\cos \frac{t}{2}+2{\sin }^2\frac{t}{2}}$

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

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

$\Rightarrow \frac{1}{\sqrt{3}+0}=\frac{1}{\sqrt{3}}$.