Assumption: $\alpha>0,\beta>0$.
I am interested in the asymptotic behavior of this ratio when both $\alpha$ and $\beta$ are scaled up by x: $$\frac{\beta^{\alpha-1}e^{-\beta}}{\Gamma(\alpha)}.$$ One can actually view this ratio as the Poisson probability: $$\frac{\beta^{\alpha-1}e^{-\beta}}{\Gamma(\alpha)}= P[Poisson(\beta)=\alpha-1].$$
Specifically, when both $\alpha$ and $\beta$ are scaled up by a factor $x$, does the following limit exist and what will be the limit: $$\lim_{x \to \infty} \frac{(\beta x)^{\alpha x-1}e^{-\beta x}}{\Gamma(\alpha x)}*x?$$
In addition, what will be the convergence rate? Can we say that $$\frac{(\beta x)^{\alpha x-1}e^{-\beta x}}{\Gamma(\alpha x)}*x = O(\sqrt{x})?$$
I tried Stirling's approximation, which gives me: $$\frac{(\beta x)^{\alpha x-1}e^{-\beta x}}{\Gamma(\alpha x)}*x=\frac{(\beta x)^{\alpha x-1}e^{-\beta x}}{\sqrt{2 \pi (\alpha x-1)} \left(\frac{\alpha x-1}{e}\right)^{\alpha x-1}}*x=\frac{(\frac{\beta x}{\alpha x -1})^{\alpha x-1}e^{\alpha x -\beta x -1}}{\sqrt{2 \pi} } *\sqrt{\frac{x}{\alpha x-1}}.$$
Is this the correct way to do it? Thank you in advance.