Approximation to x+y Choose y

Theorem

 * $\ds \lim_{x, y \mathop \to \infty} \dbinom {x + y} y = \sqrt {\dfrac 1 {2 \pi} \paren {\frac 1 x + \frac 1 y} } \paren {1 + \dfrac y x}^x \paren {1 + \dfrac x y}^y$

Proof
It can be assumed that both $x$ and $y$ are integers.