For each $\mathrm{x}\in \mathrm{R}$  let [x] denotes greatest integer  less than or equal to x then $\underset{x\to 0^{-}}{\lim }\frac{x\left(\right[x]+|x\left|\right)\sin \left[2\right]}{\left|x\right|}$  

$\underset{x\to 0^{-}}{\lim }\frac{x\left(\right[x]+|x\left|\right)\sin \left[x\right]}{\left|x\right|}$$=\underset{\mathrm{x}\to 0^{-}}{lim} \frac{\mathrm{x}(-1-\mathrm{x})\sin (-1)}{-\mathrm{x}}$

                                     $\begin{array}{c}=\underset{x\to 0^{-}}{lim} -(1+x)\sin 1\\ =-\sin 1\end{array}$