Question

The value of ${lim}_{x\to 0}{\left[\frac{(1+x{)}^{1/x}}{e}\right]}^{1/x}$ is

• A. $\sqrt{e}$
• B. $\frac{1}{\sqrt{e}}$
• C. $\frac{1}{e}$
• D. $e$

Answer 1

The correct option is B $\frac{1}{\sqrt{e}}$

$\begin{array}{c}{lim}_{x\to 0}{\left[\frac{{\left(1+x\right)}^{\frac{1}{x}}}{e}\right]}^{\frac{1}{x}}\\ {lim}_{x\to 0}\frac{\left[{\left(1+x\right)}^{\frac{1}{x}}-e\right]}{ex}=e^{{lim}_{x\to 0}\frac{e\left(1-\frac{x}{2}+...\right)-e}{ex}}\\ e^{\frac{-1}{2}}=\frac{1}{\sqrt{e}}\\ Hence,theoptionBiscorrectanswer\end{array}$