Question

$\underset{x\to \pi /3}{lim}\frac{3\tan x-{\tan }^{3}x}{\cos (x+\frac{\pi }{6})}$ is equal to

Answer 1

Reqd. limit is equal to
$=\underset{x\to \pi /3}{lim}\frac{\tan x(\sqrt{3}-\tan x)(\tan x+\sqrt{3})}{\left(\frac{1}{2}\cos x(\sqrt{3}-\tan x)\right)}$
$=\underset{x\to \pi /3}{lim}\frac{2\tan x(\tan x+\sqrt{3})}{\cos x}$
$=\frac{2\left(\sqrt{3}\right)(\sqrt{3}+\sqrt{3})}{1/2}=24$