$\frac{n!}{(n+\frac{1}{x})!} = ?$ for some positive real $x$ Is there a simple way to express : $\frac{n!}{(n+\frac{1}{x})!} = ?$ where $x, n$ are positive real numbers ? 

The answer you are looking for is found in the identity
$$\frac{\Gamma(n+x)}{\Gamma(n)}=(x)_n$$
where $(x)_n$ is the Pochhammer polynomial.
Ref: K. Oldham, J. Myland, & J. Spanier, An Atlas of Functions, Springer.
