= Exponential domination of polynomials
For every <real number> $a>0$ and nonnegative <integer> $m$, \b[$x^m e^{-ax}\to0$ 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 $x>0$, $e^{ax}\ge (ax)^{m+1}/(m+1)!$, so
$$
0\le x^m e^{-ax}\le\frac{(m+1)!}{a^{m+1}x}\longrightarrow0.
$$
Apply 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.
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