limit of variant form of incomplete Gamma function

Question

What is the limit of

$$\lim_{M\rightarrow\infty}M\frac{(Mt)^Me^{-Mt}}{M!}$$

where $0<t<\infty$, $M\in\mathbb{N}$

It is supposed to be 0 by the data observed but i could not prove it theoretically.

Answer 1

Using Stirling's formula for $M!$ we get $$ M\,\frac{(M\,t)^Me^{-Mt}}{M!}\sim\sqrt{\frac{M}{2\,\pi}}\,t^M\,e^{-Mt}. $$