Factorial of Integer plus Reciprocal of Integer

Theorem
Let $x \in \Z$ be a positive integer.

Then:
 * $\ds \lim_{n \mathop \to \infty} \dfrac {\paren {n + x}!} {n! n^x} = 1$

Proof
We have that:

As $n \to \infty$, the quantity on the indeed tends to $1$.