Задача №1998

\[ \lim_{x\to\frac{\pi}{3}}\frac{\tg^3{x}-3\tg{x}}{\cos\left(x+\frac{\pi}{6}\right)} =\left[\frac{0}{0}\right] =\lim_{x\to\frac{\pi}{3}}\frac{\tg{x}\cdot\left(\tg^2{x}-3\right)}{\cos\left(x+\frac{\pi}{6}\right)} =\lim_{x\to\frac{\pi}{3}}\left(\tg{x}\left(\tg{x}+\sqrt{3}\right)\cdot\frac{\tg{x}-\tg\frac{\pi}{3}}{\cos\left(x+\frac{\pi}{6}\right)}\right)=\\ =\lim_{x\to\frac{\pi}{3}}\left(\tg{x}\left(\tg{x}+\sqrt{3}\right)\cdot\frac{\frac{\sin\left(x-\frac{\pi}{3}\right)}{\cos{x}\cos\frac{\pi}{3}}}{\cos\left(x+\frac{\pi}{6}\right)}\right) =\lim_{x\to\frac{\pi}{3}}\left(\frac{-2\tg{x}\left(\tg{x}+\sqrt{3}\right)}{\cos{x}}\cdot\frac{\cos\left(x+\frac{\pi}{6}\right)}{\cos\left(x+\frac{\pi}{6}\right)}\right) =-24. \]