TITLE: Prove that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$
QUESTION [12 upvotes]: Using a graphing calculator, it seems that $\lim _{x\to \infty \:}(1+\frac{x^x}{x!})^{\frac{1}{x}} = e$. How can this be proven?

REPLY [6 votes]: Using the fact 

$$\lim_{n\to \infty}  a_n^{1/n}=\lim_{n\to \infty} \frac{a_{n+1}}{a_n}  $$

we have

$$\frac{a_{n+1} }{a_n}  =  \frac{ (n+1)! + (1+n)^{1+n} }{(n+1)!+(1+n)n^n} \sim_{n\sim \infty}   \frac{ (1+n)^{1+n} }{n^{n+1} } \longrightarrow_{n\to \infty} e $$