I'm trying to prove that given a continuous function $f:[0,\infty) \to \mathbb{R}$ such that $\lim_{x \to \infty} f(x)=\alpha\in \mathbb{R}$, then it can be approximated uniformly on its domain by functions of the form $P(e^{-x})$, where $P$ is a polynomial.
So far i have used the Weierstrass approximation theorem to show that for every $0<r<1$ and for every $\epsilon>0$ there exists a polynomial $P$ such that $|f(x)-P(e^{-x})|<\epsilon$ for every $x\in [r,1]$ and with this i have formed a sequence $(P_n)_{n\in \mathbb{N}}$ such that $|f(x)-P_n(e^{-x})|<1/n$ for every $n\in \mathbb{N}$. Any suggestions on how to go forward?
If $\lim_{x \rightarrow \infty} f(x) = 0$ then it is possible to approximate it by a subalgebra since $[0,\infty)$ is locally compact.
Taken from https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem
The same argument works even if $\lim_{x \rightarrow \infty} f(x) = b$ for some constant $b < \infty$.