$\lim _{x ightarrow 0} \frac{x an 2 x-2 x an x}{(1-\cos 2 x)^2}$ is

Question

$\lim _{x ightarrow 0} \frac{x 	an 2 x-2 x 	an x}{(1-\cos 2 x)^2}$ is

Aicharya · Accepted Answer

$\begin{aligned} & \lim _{x ightarrow 0} \frac{x 	an 2 x-2 x 	an x}{(1-\cos 2 x)^2} \ & =\lim _{x ightarrow 0} \frac{x(	an 2 x-2 	an x)}{\left(2 \sin ^2 xight)^2}\end{aligned}$ $\begin{aligned} & =\lim _{x ightarrow 0} \frac{x(	an 2 x-2 	an x)}{4 \sin ^4 x} \ & \left.=\frac{1}{4} \lim _{x ightarrow 0} \frac{\left\{\left(2 x+\frac{1}{3}(2 x)^3+\frac{2}{15}(2 x)^5+\ldotsight)ight.}{-2\left(x+\frac{x^3}{3}+\frac{2}{15} x^5+\ldotsight)}ight\} \ & =\frac{1}{4}\left(\frac{8}{3}-\frac{2}{3}ight)=\frac{2}{4}=\frac{1}{2}\end{aligned}$

$\lim _{x \rightarrow 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^2}$ is

Solution