Mathematics - Continuity - Differentiability Question with Solution | TestHub

$f(0)=0;f(x)=x{e}^{-\left(\frac{1}{|x|}+\frac{1}{x}\right)}$ R.H.L. ${\lim }_{h\to 0}(0+h){\mathrm{e}}^{-2/h}={\lim }_{h\to 0}\frac{h}{{\mathrm{e}}^{2/h}}=0$ L.H.L ${\mathrm{Lim}}_{h\to 0}(0-h)e^{-\left(\frac{1}{h}-\frac{1}{h}\right)}=0$ Therefore, $f(x)$ is continuous R.H.D. ${\lim }_{h\to 0}\frac{(0+h)e^{-\left(\frac{1}{h}+\frac{1}{h}\right)}-h{e}^{-\left(\frac{1}{h}+\frac{1}{h}\right)}}{h}=0$ L.H.D. ${\mathrm{Lim}}_{h\to 0}\frac{(0-h){\mathrm{e}}^{-\left(\frac{1}{h}-\frac{1}{h}\right)}-h{e}^{-\left(\frac{1}{h}+\frac{1}{h}\right)}}{-h}=1$ Therefore, L.H.D. $\ne$ R.H.D. $\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=0$.