Question

If $f\left(x\right)=x\sin \frac{1}{x},x\ne 0,$ then the value of the function at x = 0, so that the function is continuous at x = 0, is
(a) 0
(b) −1
(c) 1
(d) indeterminate

Answer 1

(a) 0

Given: $f\left(x\right)=x\sin \frac{1}{x},x\ne 0$

Here,
$\underset{x\to 0}{\lim }x\sin \left(\frac{1}{x}\right)=\underset{x\to 0}{\lim }x\underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)=0\times \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)=0$


If f(x) is continuous at x = 0, then$\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$.

$\Rightarrow f\left(0\right)=0$