I am reading up on the gamma function and have seen a formula that I can't connect to the usual integral definition. Namely,
$$ \Gamma(x) = \lim_{n\rightarrow \infty}\frac{n!n^{x-1}}{x(x+1)\cdots(x+n-1)}, \qquad x\neq 0,-1,-2,\dots $$
How can I connect this formula with the standard definition:
$$ \Gamma(x) = \int^\infty_0 t^{x-1}e^{-t}dt $$