Question

If $f(x)=\{\begin{array}{ll}x\sin \frac{1}{x},& x\ne 0\\ k,& x=0\end{array}$ is continuous at $x=0$ , then the value of $k$ will be

• A. 1
• B. -1
• C. 0
• D. None of these

Answer 1

Answer: (C)

Given that $f(x)=\{\begin{array}{ll}x\sin \frac{1}{x},& x\ne 0\\ k& ,x=0\end{array}$
$LHL=\underset{x\to 0^{-}}{\lim }f(x)$
$=\underset{x\to 0^{-}}{\lim }x\sin \frac{1}{x}$
$=\underset{h\to 0}{\lim }(0-h)\sin \frac{1}{(0-h)}$
$=\underset{h\to 0}{\lim }h\sin \frac{1}{h}=0$
Since, $f(x)$ is continuous at $x=0$
$\therefore LHL=f(0)$
$\Rightarrow 0=k$