Help with a limit question

Question

The question is: How do we evaluate $$\lim_{n\rightarrow \infty}\frac{n!n^{x}}{x(x+1)...(x+n)}(Re(x)>0)$$ Any hint will be appreciated.

Answer 1

Let $a_{n}$ be your sequence and set $b_{n}=1/a_{n}$. Apply the ration test for convergence of series and show that |b_{n+1}/b_{n}| converges to 0 as $n$ goes to infinity, and conclude that the series of general term $b_{n}$ is convergent. Thus the general terms of the series, i.e., the sequence $b_{n}$, approaches 0. Form there conclude that your sequence (in absolute value) goes to infinity