Question:
For each $\mathrm{x} \varepsilon \mathrm{R}$, let $[\mathrm{x}]$ be the greatest integer less than or equal to $x$. Then
$\lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{|x|}$ is equal to
Correct Option: 1
Solution:
$\lim _{x \rightarrow 0^{-}} \frac{x([x]+|x|) \sin [x]}{|x|}$
$\mathrm{x} \rightarrow 0^{-}$
$[x]=-1 \Rightarrow \lim _{x \rightarrow 0^{-}} \frac{x(-x-1) \sin (-1)}{-x}=-\sin 1$
$|x|=-x$