$\underset{x\to -1}{lim} \frac{\cos 2-\cos 2x}{x^2-\left|x\right|}$ is equal to 

Solution:

We have,

$\begin{array}{l}\underset{x\to -1}{lim} \frac{\cos 2-\cos 2x}{x^2-\left|x\right|}\\ =\underset{x\to -1}{lim} \frac{2\sin (1+x)\sin (x-1)}{x^2+x}[\because |x|=-x\text{ for }x<0]\\ =2\underset{x\to -1}{lim} \frac{\sin (x+1)}{x+1}\times \frac{\sin (x-1)}{x}\\ =2\times 1\times \frac{\sin (-2)}{(-1)}=2\sin 2.\end{array}$