Question

If $\left\{x\right\}$ denotes fractional part of $x,$ then the value of $\underset{x\to 1}{lim}\frac{x\sin \left\{x\right\}}{x-1}$ is

• A. $0$
• B. $-1$
• C. $1$
• D. It does not exist

Answer 1

The correct option is D It does not exist

$\underset{x\to 1}{lim}\frac{x\sin \left\{x\right\}}{x-1}$
$R.H.L.=\underset{h\to 0}{lim}\frac{(1+h)\sin \{1+h\}}{h}$
$=\underset{h\to 0}{lim}\frac{(1+h)\sin h}{h}=1$
$L.H.L.=\underset{h\to 0}{lim}\frac{(1-h)\sin \{1-h\}}{-h}$
$=\underset{h\to 0}{lim}\frac{(1-h)\sin (1-h)}{-h}=-\infty$
Therefore, Limit does not exist.