How can one prove that for any $x>1$: $$\lim\limits_{n\to \infty }\left(\frac{n^x \Gamma \left(n+1\right)}{\Gamma \left(x+n+1\right)}\right)=1$$
It is easy to show this for $x$ a natural number since we can write the gamma function as a factorial but I would like some ideas on how to prove this for $x$ not a natural number.
