Question

Solve

$\underset{n\to \infty }{lim}{\left(1+\frac{1}{2n}\right)}^n$

Answer 1

$L={lim}_{n\to \infty }{(1+\frac{1}{2n})}^n\log l={lim}_{n\to \infty }\log (1+\frac{1}{2n})let\frac{1}{n}=t,n\to \infty \Rightarrow t\to 0\log l={lim}_{t\to 0}\frac{\log (1+0.5t)}{t}\left[\frac{0}{0}form\right]$

Using L-Hopital rule.

$\log l=\frac{1}{1+0.5t}\times 0.5\log l=\frac{1}{2}l=e^{\frac{1}{2}}Limit=\sqrt{e}$