$\mathop {\lim }\limits_{x \to 0} \frac{x}{|x| + {x^2}} = $

The value of $\lim _{x}$ ${\rightarrow \frac{\pi}{2}} \frac{\left(1-\tan \left(\frac{x}{2}\right)\right)(1-\sin x)}{\left(1+\tan \left(\frac{x}{2}\right)\right)(\pi-2 x)^3}$ is

$\mathop {\lim }\limits_{x \to 0} \cos \frac{1}{x}$

For each $x \in \mathbb{R}$,let $[x]$ be the greatest integer less than or equal to $x$. Then $\lim_{x \to 0^+} \frac{x([x] + |x|) \sin [x]}{|x|}$ is equal to

Find the limit: $\mathop {\lim }\limits_{x \to 2} \left[\frac{x^{3}-4 x^{2}+4 x}{x^{2}-4}\right]$

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} (\text{cosec}\,x - \cot x)$