Limit $\lim_{x\to \frac{\pi}{3}} \frac{\sin(x-\frac{\pi}{3})}{1-2 \cos{x}}$ I need to evaluate following limit:
$$\lim_{x\to \frac{\pi}{3}} \frac{\sin(x-\frac{\pi}{3})}{1-2 \cos{x}}$$
Tried multiplying with the argument inside the sinus function but finished with this limit:
$$\lim_{x\to \frac{\pi}{3}} \frac{x - \frac{\pi}{3}}{1-2\cos{x}}$$
I know this limit is $\frac{1}{\sqrt{3}}$, but I have no steps which would show how I obtained this value
PS: l'Hospital is forbidden

$$ \frac {\sin(x-\frac{\pi}{3})} {1-2 \cos{x}} = \frac {\sin(x-\frac{\pi}{3})} {x-\frac{\pi}{3}}  \frac {x-\frac{\pi}{3}} {1-2 \cos{x}}=\frac {\sin y} {y}  \frac {{y}} {1-2 \cos{\left(y+\frac{\pi}{3}\right)}}=\frac {\sin y} {y}  \frac {{y}} {1-\cos y+\sqrt{3}\sin y}\to \frac{1}{\sqrt{3}}     $$
