<s>[INST] Show that $\lim_{n\rightarrow \infty}n \big[1-\frac{(n+1)^{n}}{en^{n}}\big]=\frac{1}{2}$ This is exercise from Guidorizzi's book Cálculo (In Portuguese).
Show that $\lim_{n\rightarrow \infty}n \big[1-\frac{(n+1)^{n}}{en^{n}}\big]=\frac{1}{2}.$
All I managed to do is rewrite the equation as $n \big[1-\frac{(1+\frac{1}{n})^{n}}{e}\big]$. 
What to do from now on?
 [/INST] $$\log\left(1 + \frac{1}{n}\right)^n = n \log \left(1+\frac{1}{n}\right) = n\left(\frac{1}{n}-\frac{1}{2n^2} + O\left(\frac{1}{n^3}\right)\right)$$
Thus 
$$\left(1 + \frac{1}{n}\right)^n =e^{1 - \frac{1}{2n} + O\left(\frac{1}{n^2}\right)} $$
Substituting, we need to find 
$$\lim_{n \to \infty} n\left(1 - \frac{e^{1 - \frac{1}{2n} + O\left(\frac{1}{n^2}\right)}}{e}\right) = n\left(1-e^{\frac{-1}{2n} + O\left(\frac{1}{n^2}\right)}\right)$$
Now using the expansion of $e^x$, 
you get
$$n\left(1-e^{\frac{-1}{2n} + O\left(\frac{1}{n^2}\right)}\right) = n\left(\frac{1}{2n} + O\left(\frac{1}{n^2}\right)\right)$$
which tends to $\frac{1}{2}$ and $n \to \infty$.
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