Question

$\begin{array}{ccc}f\left(x\right)=(1+2x{)}^{1/x},& for& x\ne 0\\ =e^2,& for& x=0\end{array}\}$ at $x=0$.Find whether $f\left(x\right)$ is continuous or not

Answer 1

For $x\ne 0,f\left(x\right)=(1+2x{)}^{1/x}$

At $x=0,f\left(0\right)=\underset{x\to 0}{lim}(1+2x{)}^{1/x}=1^{\infty }=$ undetermined form

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

So f(x) is continuous