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. $0$
• C. $-\sin 1$
• D. $1$

Answer 1

The correct option is C $-\sin 1$

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

$=\underset{h\to 0}{lim}-\frac{(1-h)\sin 1}{1}$
$=-\sin 1$