Gaussian decay dominates every power
= Gaussian decay dominates every power
{c}
{title2=$|t|^m e^{-t^2}\to0\quad(|t|\to\infty)$}
For every fixed $m\ge0$, choose an integer $N>m/2$. The positive <exponential series> gives $e^{t^2}\ge t^{2N}/N!$, so $|t|^m e^{-t^2}\le N!|t|^{m-2N}\to0$. Equivalently, $|x|^{-m}e^{-1/x^2}\to0$ as $x\to0$.