TITLE: Find the limit of $(1+\frac{1}{n})^{n^2}/e^n$ without using derivatives
QUESTION [6 upvotes]: Try to use just basic operations with limits to find this one: 
$$\lim_{n\to\infty}\frac{(1+\frac{1}{n})^{n^2}}{e^n}$$
I have some ideas. Is it possible that they are right?
$$\lim_{n\to\infty}\frac{(1+\frac{1}{n})^{n^2}}{e^n}=\lim_{n\to\infty}\frac{((1+\frac{1}{n})^{n})^n}{e^n}=\lim_{n\to\infty}\frac{e^{n}}{e^n} = 1$$

REPLY [4 votes]: Since
\begin{eqnarray}
\ln\left(1+\frac1n\right)^{n^2}&=&n^2
\ln\left(1+\frac1n\right)=n^2\left[\frac1n-\frac{1}{2n^2}+\frac{1}{3n^3}+o\left(\frac{1}{n^3}\right)\right]\\
&=&n-\frac12+\frac{1}{3n}+o\left(\frac{1}{n}\right),
\end{eqnarray}
we have
$$
\lim_{n\to\infty}\frac{\left(1+\frac1n\right)^{n^2}}{e^n}=
\lim_{n\to\infty}\exp\left[-\frac12+\frac{1}{3n}+o\left(\frac1n\right)\right]=e^{-\frac12}.
$$