Задача №1997

\[ \lim_{x\to\frac{\pi}{3}}\frac{\sin\left(x-\frac{\pi}{3}\right)}{1-2\cos{x}} =\left[\frac{0}{0}\right] =\lim_{x\to\frac{\pi}{3}}\frac{\sin\left(2\cdot\left(\frac{x}{2}-\frac{\pi}{6}\right)\right)}{2\cdot\left(\cos\frac{\pi}{3}-\cos{x}\right)}=\\ =\lim_{x\to\frac{\pi}{3}}\frac{2\sin\left(\frac{x}{2}-\frac{\pi}{6}\right)\cos\left(\frac{x}{2}-\frac{\pi}{6}\right)}{4\sin\left(\frac{x}{2}+\frac{\pi}{6}\right)\sin\left(\frac{x}{2}-\frac{\pi}{6}\right)} =\lim_{x\to\frac{\pi}{3}}\frac{\cos\left(\frac{x}{2}-\frac{\pi}{6}\right)}{2\sin\left(\frac{x}{2}+\frac{\pi}{6}\right)} =\frac{1}{\sqrt{3}}. \]