For each  $t\in R$, let [t] be the greatest integer less than or equal to t. Then,$\underset{x\to 1^{+}}{\lim }\frac{\left(1-\left|x\right|+\sin \left|1-x\right|\right)\sin \left(\frac{\pi }{2}\left[1-x\right]\right)}{\left|1-x\right|\left[1-x\right]}$
   
 

$\begin{array}{l}{\mathrm{Lim}}_{x\to 1^{+}}\frac{(1-|x|+\sin |1-x\left|\right)\sin \left([1-x]\frac{\pi }{2}\right)}{|1-x|[1-x]}\\ ={\mathrm{Lim}}_{x\to 1^{+}}\frac{(1-x+\sin (x-1\left)\right)\sin \left(-\frac{\pi }{2}\right)}{(x-1)(-1)}={\mathrm{Lim}}_{x\to 1^{+}}\frac{-(x-1)+\sin (x-1)}{(x-1)}=-1+1=0\end{array}$