Can't find a seemingly simple limit $\lim_{n\to\infty}\frac{(n+k)!}{n^n}$

Apply the ratio test to the series $$ \sum_{n=0}^{\infty}\frac{(k+n)!}{n^n} $$ The ratios to evaluate are $$ \frac{(k+n+1)!}{(n+1)^{n+1}}\frac{n^n}{(k+n)!}=\frac{k+n+1}{n+1}\frac{n^n}{(n+1)^n} $$ Note that the limit of the first fraction is $1$ and the limit of the second fraction is $1/e<1$.

By the ratio test, the series is convergent. Hence $$ \lim_{n\to\infty}\frac{(k+n)!}{n^n}=0 $$