I need to show that the $$\lim_{t\to\infty} t^m e^{{\alpha}t}=0$$ where $m \in \mathbb N_0$ $\Re (\alpha)<0$ and
$$ \max_{0 \leq t< \infty}\left|t^m\cdot e^{{\alpha}t}\right|= \left(\frac{m}{-\Re(\alpha)}\right)^m\cdot e^{-m}$$ $m \in \mathbb N_0$, $\Re (\alpha)<0$ by using L'Hopital's rule.
How do I start? Thank you so much! Klara