TITLE: What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?
QUESTION [7 upvotes]: How to solve the following limit question?
$$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$
Thanks a lot.

REPLY [2 votes]: \begin{eqnarray*}
\displaystyle &&\lim_{n\rightarrow\infty}(1+\frac{1}{n})^{n^2}\frac{1}{e^n}\\
 &=&\displaystyle e^{n^2ln(1+\frac{1}{n})-n}\\
 &=&e^{\displaystyle \lim_{n\rightarrow\infty}\left(n^2ln(1+\frac{1}{n})-n\right)}\\
 &=&e^{\displaystyle \lim_{m\rightarrow0^+}\left(\left(\frac{1}{m}\right)^2ln(1+m)-\frac{1}{m}\right)}\\
 &=&e^{\displaystyle \lim_{m\rightarrow0^+}\left(\frac{ln(1+m)-m}{m^2}\right)}\\
 &=&e^{\displaystyle \lim_{m\rightarrow0^+}\frac{\frac{1}{1+m}-1}{2m}}\\
 &=&e^{\displaystyle \lim_{m\rightarrow0^+}\frac{-m}{2m(1+m)}}\\
 &=&e^{\displaystyle \lim_{m\rightarrow0^+}\frac{-1}{2(1+m)}}\\
&=&e^{-\frac{1}{2}} 
\end{eqnarray*}