gamma function limit involving stirlings formula

Hello, I am trying to show the following: $\displaystyle \lim_{x\rightarrow\infty}\frac{\Gamma (x+c)}{x^c \Gamma (x)}=1$, where $\displaystyle c\in\mathbb{R}$. Recall Stirling's formula: $\displaystyle \lim_{x\rightarrow\infty}\frac{\Gamma (x+1)}{(x/e)^x \sqrt{2\pi x}}=1$. Any suggestions on how to prove the former limit using this formula? I have tried the change of variables $\displaystyle t=x(c+u)$, but got stuck. Any help would be greatly appreciated.

Thanks in advance,

Arlington