The value of $\mathop {\lim }\limits_{x \to {0^ + }} {x^m}{(\log x)^n}$,where $m, n \in N$,is

$\lim _{x \rightarrow 0} \frac{2 \sin x-\sin 2 x}{x^3}$ is equal to

Evaluate $\mathop {\lim }\limits_{x \to {1^ + }} \frac{{{{\left( {1 + \left\{ x \right\}} \right)}^{\frac{1}{{\left\{ x \right\}}}}} - \frac{e}{{\sqrt {{e^{\left\{ x \right\}}}} }}}}{{1 - \cos \left\{ x \right\}}}$ (where $\{.\}$ denotes the fractional part function).

$\lim _{x \rightarrow 0} \frac{\log _{e}(1+x)}{3^{x}-1}$ is equal to

Let $[x]$ denote the greatest integer less than or equal to $x$ and $k \geq 2$ be an integer. Then $\lim_{x \rightarrow k} \frac{\sin \left(2 \pi\left([x]-\left[\frac{x}{k}\right]\right)-x\right)+\sin k}{x-k} = $

$\mathop {\lim }\limits_{x \to \pi /2} \frac{2x - \pi}{\cos x} = $