Задача №1129
Условие
Найти предел \(\lim_{x\to\frac{\pi}{6}}\frac{\sin\left(x-\frac{\pi}{6}\right)}{\frac{\sqrt{3}}{2}-\cos{x}}\).
Решение
\[
\lim_{x\to\frac{\pi}{6}}\frac{\sin\left(x-\frac{\pi}{6}\right)}{\cos\frac{\pi}{6}-\cos{x}}
=\lim_{x\to\frac{\pi}{6}}\frac{2\sin\frac{x-\frac{\pi}{6}}{2}\cos\frac{x-\frac{\pi}{6}}{2}}{2\sin\frac{x-\frac{\pi}{6}}{2}\sin\frac{x+\frac{\pi}{6}}{2}}
=\lim_{x\to\frac{\pi}{6}}\frac{\cos\frac{x-\frac{\pi}{6}}{2}}{\sin\frac{x+\frac{\pi}{6}}{2}}
=2.
\]
Ответ:
2