Bound for $e^{-\alpha x}$

Question

For a part of my proof I need to establish that $e^{-\alpha x} \lt h(x)$, where $\alpha,x \gt 0, $ and $x,\alpha \in\mathbb{R}$. I thought for a while and couldn't find a function independent of $\alpha$ that fulfills my criteria. Any ideas?

Answer 1

Also note that: $$e^{-\alpha x}=\sum_{n=0}^\infty\frac{(-1)^n\alpha^nx^n}{n!}$$ A polynomial of this expansion could be used, although take care to see that it is consistently greater as this is a series with alternating sign