NIMCET 2022 — Mathematics PYQ

(b) The function $f(x)=\{\begin{array}{ll}(1+2x{)}^{1/x},& amp;x\ne 0\\ e^2,& amp;x=0\end{array}$

Check continuity at $x=0$

LHL = RHL = $f(0)$

$\Rightarrow {\lim }_{x\to 0}(1+2x{)}^{1/x}=e^2$

$\Rightarrow e^{{\lim }_{x\to 0}(1+2x-1)\times \frac{1}{x}}=e^2$

$[\text{Use formula }{\lim }_{x\to a}(1+f(x){)}^{g(x)}$

${\lim }_{x\to 0}(1+f(x)-1)g(x)$

$\text{then, }{e}^{x\to 0}]$

$\Rightarrow {\lim }_{x\to 0}\frac{2x}{x}=e^2$

$\Rightarrow e^2=e^2$

$f(x)$ is continuous at $x=0$.

For differentiability:

$f(x)=(1+2x{)}^{1/x}$

$\log f(x)=\frac{1}{x}\log (1+2x)$

Differentiate both side

$\frac{1}{f(x)}{f}^{\prime }(x)=\frac{2}{x(1+2x)}+\log (1+2x)\left(\frac{-1}{x^2}\right)$

$f^{\prime }(x)=(1+2x{)}^{1/x}\left[\frac{2}{x(1+2x)}-\frac{\log (1+2x)}{x^2}\right]$

$f^{\prime }(x)$ is undefined at $x=0$.

$\Rightarrow f(x)$ is not differentiable at $x=0$