$f\left(x\right)=\left\{\begin{array}{cc}{x}^2\left(\frac{e^{1/x}-e^{-1/x}}{e^{1/x}+e^{-1/x}}\right),& x\ne 0\\ 0,& x=0\end{array}\right.$    Then

$$\begin{gathered} At x=0 \\ \underset{x\to 0^{-}}{lim} f\left(x\right)=\underset{h\to 0}{lim} h^2\left(\frac{e^{-1/h}-e^{1/h}}{e^{-1/h}+e^{1/h}}\right)=0 \end{gathered}$$

$\underset{x\to 0+}{lim} f\left(x\right)=\underset{h\to 0}{lim} h^2\left(\frac{e^{1/h}-e^{-1/h}}{e^{1/h}+e^{-1/h}}\right)=0$

And $f\left(0\right)=0$, $f\left(x\right)$ is continuous at $x=0.$

Also  $\text{ L.H.D. }=\underset{h\to 0}{lim} h\frac{e^{2/h}-1}{e^{2/h}+1}=0$ and

$\text{ R.H.D. }=\underset{h\to 0}{lim} \frac{1-e^{-2/h}}{1+e^{2/h}}=0$