Question

Determine if f defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2\sin \frac{1}{x},& ifx\ne 0\\ 0,& ifx=0\end{array}$ is a continuous function?

Answer 1

We have given

$f\left(x\right)=\{\begin{array}{c}{x}^2\sin \frac{1}{x},ifx\ne 0\\ 0,ifx=0\end{array}$

we know that $f\left(0\right)=0$

We have to determine if $f$ is a continuous function

$f\left(x\right)=x^2\sin \frac{1}{x}$ at $x\ne 0$

Find the limit of $f\left(x\right)$ at $x=0$

$R.H.L=\underset{x\to (0+h)}{lim}{x}^2\sin \frac{1}{x}=\underset{h\to 0}{lim}(0+h{)}^2\sin \frac{1}{0+h}$

$=\underset{h\to 0}{lim}{h}^2\sin \frac{1}{h}=0^2\times \sin \infty$

$L.H.L=\underset{x\to (0-h)}{lim}{x}^2\sin \frac{1}{x}=\underset{x\to 0}{lim}(-h{)}^2\sin \frac{1}{0-h}$

$L.H.L=\underset{h\to 0}{lim}=-h^2\sin \frac{1}{h}=-0^2\sin \infty =0$

$R.H.S=\underset{x\to 0}{lim}0=0$

$L.H.L=R.H.L=f\left(0\right)$

So, the function $f\left(x\right)$ is continuous function