A limit using the Euler number: $\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$ What is answer of this limit and how can I get it? $c$ and $i$ are constants.
$$\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$$
I guess it will envolve some Neper/the Euler number $e$. I tried to rearrange terms of factorial and exponents in a good way but I couldn't make any conclusion so I think couldn't find the nice shape of writing this expression.

Hint. We assume $c,i$ are any fixed real numbers. 
By setting $\displaystyle u_n=\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$, one gets, as $n \to \infty$,
$$
\begin{align}
\frac{u_{n+1}}{u_n}&=\frac{(n+1)!}{(n+1-i)!}\left(\frac{c}{n+1}\right)^{n+1-i}\cdot \frac{(n-i)!}{n!}\left(\frac{n}{c}\right)^{n-i}
\\\\&=\frac{(n+1)}{(n+1-i)}\cdot \frac{c}{n+1}\cdot \left(1+\frac1n \right)^{-n+i}
\\\\& \sim \frac{c}{e(n+1)}
\\\\& \to 0.
\end{align}
$$
Can you take it from here?
