Let $f\left(x\right)=\left\{\begin{array}{l}x{e}^{-\left(\frac{1}{\left|x\right|}+\frac{1}{x}\right)} ; x\ne 0 \\ 0 ; x=0\end{array}\right.$then which of the following is true?

The given function is $f\left(x\right)=\left\{\begin{array}{l}x{e}^{-\left(\frac{1}{\left|x\right|}+\frac{1}{x}\right)} ; x\ne 0\\ 0 ; x=0\end{array}\right.$

Consider both left hand limit and right hand limit at $x=0$

Both are equal and equal to $f\left(0\right)$

Hence, the function is continuous at $x=0$

But the function is not differentiable at $x=0$ because left hand derivative and right hand derivative are not equal at $x=0$

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

Therefore, the given function is $\boxed{continuos but not differentiable at x=0}$