\(L=\lim\limits_{x\rightarrow\frac{\pi}{3}}\frac{\tan^3x-3\tan x}{\cos\left(x+\frac{\pi}{6}\right)}=\lim\limits_{x\rightarrow\frac{\pi}{3}}\frac{\tan x\left(\tan^2x-3\right)}{\cos\left(x+\frac{\pi}{6}\right)}\)
\(=\sqrt{3}\lim\limits_{x\rightarrow\frac{\pi}{3}}\frac{\left(\tan x-\sqrt{3}\right)\left(\tan x+\sqrt{3}\right)}{\sin\left(\frac{\pi}{3}-x\right)}=\sqrt{3}.2\sqrt{3}\lim\limits_{x\rightarrow\frac{\pi}{3}}\frac{\tan x-\sqrt{3}}{\sin\left(\frac{\pi}{3}-x\right)}\)
\(=6\lim\limits_{x\rightarrow\frac{\pi}{3}}\frac{\sin\left(\frac{\pi}{3}-x\right)}{\cos x.\cos\frac{\pi}{3}\sin\left(\frac{\pi}{3}-x\right)}=-12\lim\limits_{x\rightarrow\frac{\pi}{3}}\frac{1}{\cos x}=-24\)
