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Let $f(x)= \begin{cases}\frac{5 e^{1 / x}+2}{3-e^{1 / x}}, & x \neq 0 \\ 0, & x=0\end{cases}$ Then at $x=0, x f(x)$ and $f(x)$ are respectively
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Continuous and not differentiable
$f(x)=\left\{\begin{array}{cc}\frac{5 e^{1 / x}+2}{3-e^{1 / x}} & x \neq 0 \\ 0 & x=0\end{array}\right.$
Then $x f(x)=\left\{\begin{array}{cc}\frac{x\left(5 e^{1 / x}+2\right)}{3-e^{1 / x}} & x \neq 0 \\ 0 & x=0\end{array}\right.$
Let us check continuity of $x f(x)$ at $x=0$
$\begin{aligned}
& \text { L.H.L }=\lim _{h \rightarrow 0} \frac{(-h)\left[5 e^{-1 / h}+2\right]}{3-e^{-1 / h}}=0 \\
& \text { R.H.S }=\lim _{h \rightarrow 0} \frac{\left[55^{\frac{1}{h}}+2\right]}{3-e^{\frac{1}{h}}}=0
\end{aligned}$
$x f(x)=0$
$\therefore \quad x f(x)$ is continous on $x=0$
Let us check differentiability of $f(x)$ at $x=0$ :
$\begin{aligned}
& \text { L.H.D. }=\lim _{h \rightarrow 0} \frac{f(a-h)-\mathrm{f}(0)}{-h} \\
& =\lim _{h \rightarrow 0} \frac{\frac{5 e^{-1 / n}+2}{3-e^{-1 / n}-0}}{-h}
\end{aligned}$
$=\lim _{h \rightarrow 0}-\frac{1}{h}\left(\frac{5 e^{-1 / n}+2}{3-e^{-1 / h}}\right) \rightarrow$
$\because \quad$ L.H.D is not finite at $x=0$
$\therefore \quad f(x)$ is not differentiable at $x=0$
Then $x f(x)=\left\{\begin{array}{cc}\frac{x\left(5 e^{1 / x}+2\right)}{3-e^{1 / x}} & x \neq 0 \\ 0 & x=0\end{array}\right.$
Let us check continuity of $x f(x)$ at $x=0$
$\begin{aligned}
& \text { L.H.L }=\lim _{h \rightarrow 0} \frac{(-h)\left[5 e^{-1 / h}+2\right]}{3-e^{-1 / h}}=0 \\
& \text { R.H.S }=\lim _{h \rightarrow 0} \frac{\left[55^{\frac{1}{h}}+2\right]}{3-e^{\frac{1}{h}}}=0
\end{aligned}$
$x f(x)=0$
$\therefore \quad x f(x)$ is continous on $x=0$
Let us check differentiability of $f(x)$ at $x=0$ :
$\begin{aligned}
& \text { L.H.D. }=\lim _{h \rightarrow 0} \frac{f(a-h)-\mathrm{f}(0)}{-h} \\
& =\lim _{h \rightarrow 0} \frac{\frac{5 e^{-1 / n}+2}{3-e^{-1 / n}-0}}{-h}
\end{aligned}$
$=\lim _{h \rightarrow 0}-\frac{1}{h}\left(\frac{5 e^{-1 / n}+2}{3-e^{-1 / h}}\right) \rightarrow$
$\because \quad$ L.H.D is not finite at $x=0$
$\therefore \quad f(x)$ is not differentiable at $x=0$
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