How to find $\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}$ What method should I use for this limit?
$$
\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}
$$
I tried ratio test but I ended with the ugly answer 
$$\lim_{n\to \infty}\frac{(n+1)^{n-1}}{n^{n-1}}
$$ 
which would go to 1? Which means we cannot use ratio test. I do not know how else I could find this limit.

Note that for $n\ge 2$ the top is equal to $n\cdot n^{n-2}$.
The bottom is $(1)(2)\left[(3)(4)\cdots(n)\right]$. The product $[(3)(4)\cdots(n)]$ consists of $n-2$ terms, all $\le n$. So the bottom is $\le (1)(2)n^{n-2}$.
It follows that for $n\ge 2$ we have
$$\frac{n^{n-1}}{n!}\ge \frac{n\cdot n^{n-2}}{(1)(2)n^{n-2}}=\frac{n}{2}.$$
Since $\frac{n}{2}\to\infty$ as $n\to\infty$, it follows that our sequence diverges to $\infty$.
