 Exponential domination of polynomials

For every real number $a>0$ and nonnegative integer $m$, $x^m{e}^{-ax}\to 0$ as $x\to \infty$. Thus every fixed polynomial grows more slowly in magnitude than a positive-rate exponential function, regardless of the polynomial's coefficients.

A short proof comes from the power series of the exponential function. For $x>0$, $e^{ax}\ge (ax{)}^{m+1}/(m+1)!$, soApply this estimate term by term to a polynomial. For example, $x^{100}/e^x$ eventually tends to zero even though it can be extremely large at moderately sized $x$. This is an asymptotic statement, not a claim that the exponential function is larger at every positive input. It explains why exponential tails have finite moments of every fixed order and why the proof that the exponential function is a transcendental function works.