Question

The value of $\underset{x\to 0^{-}}{lim}\frac{x\left(\right[x]+|x|)\sin \left[x\right]}{|x|}$ is
(where $[.]$ is greatest integer function)

• A. $-\sin 1$
• B. $\sin 1$
• C. $0$
• D. $1$

Answer 1

The correct option is A $-\sin 1$

$L=\underset{x\to 0{}^{-}}{lim}\frac{x\left(\right[x]+|x|)\sin \left[x\right]}{|x|}$
We know,
$x\to 0^{-}\Rightarrow \left[x\right]=-1,|x|=-x$
Now,
$L=\underset{x\to 0^{-}}{lim}\frac{x(-1-x)\sin (-1)}{-x}=\underset{x\to 0^{-}}{lim}-\frac{(1+x)\sin 1}{1}=-\sin 1$