$\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$
It's not a "solution" (proof?) but I would like to give some details to your question.
$$ \vert f(x)-e^{-x}p(x)\vert=e^{-x}\vert g(x)-p(x)\vert, $$ where $\lim_{x\to\infty}e^{-x}g(x)=0$. Your question is about the density of polynomials in the weighted space (weight function is $e^{-x}$) on the half-line. More info can be found Stone-Weierstrass theorem (locally compact version). The literature of weighted approximation in $L_p$ spaces is huge.