Question

If $f\left(x\right)=x\sin \left(\frac{1}{x}\right),x\ne 0$ is continuous at $x=0,$ then the value of $f\left(0\right)$ is

• A. $1$
• B. $0$
• C. $2$
• D. $-1$

Answer 1

The correct option is B $0$

Given : $f\left(x\right)=x\sin \left(\frac{1}{x}\right),x\ne 0$
At $x=0,$
$L.H.L.=\underset{h\to 0}{lim}f(0-h)=\underset{h\to 0}{lim}(-h)\sin \left(\frac{-1}{h}\right)$
$=0\times \text{finite}=0$
Since $f\left(x\right)$ is continuous at $x=0,$
$\Rightarrow L.H.L.=f\left(0\right)=0$
$\therefore f\left(0\right)=0$