Question

Evaluate :
$\underset{x\to 0}{lim}\frac{x\ln (1+2\tan x)}{1-\cos x}$

Answer 1

$\underset{x\to 0}{lim}\frac{x\ln (1+2\tan x)}{1-\cos x}$
$=\underset{x\to 0}{lim}\frac{x\ln (1+2\tan x)}{(1-\cos x)\times 2\tan x}\times$ $2\tan x$
$=\underset{x\to 0}{lim}\frac{x\times 2\tan x}{1-\cos x}$
$=\underset{x\to 0}{lim}x.2.\frac{\tan x}{x}.\frac{x}{2{\sin }^2\frac{x}{2}}$
$=\underset{x\to 0}{lim}\frac{x^2}{{\sin }^2\frac{x}{2}}$
$=\underset{x\to 0}{lim}4\frac{{\left(\frac{x}{2}\right)}^2}{{\sin }^2\frac{x}{2}}$
$=4$.