I'm sorry if this is a simple question, but this page on Wolfram Research states that it follows from Stirling's formula that:
$ \frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta $
for large $x$, but I'm just not seeing a simple derivation. Could you help me see how this follows (other resources I've seen state that the more general formulation provided on that link, that
$\frac{\Gamma(x+\beta)}{\Gamma(x+\alpha)} \approx x^{\beta-\alpha}$
follows from some algebra using Stirling's formula, but again, I can't come up with it.