Csdrhrt

Suppose $fin mathcal{C}([1,infty))$ and $underset{xrightarrow +infty}{lim} f(x)=a$. Show that $f$ can be uniformly approximated on $[1,infty)$ by functions of the form $g(x)=p(1/x)$, where $p$ is a polynomial.

So I know I need to use the Weierstrass theorem, but I do not know if my approach is right. First, I know that Weierstrass is applied only on intervals of the form $[a,b]$. Would I be able to use it on the interval $[a, a+n]$ and show I can approximate $f$ for any $nin mathbb{N}$?

I am also thinking that the end goal here is for my polynomial, $p_n$, to equal $a$ at $x=0$? That way $underset{xrightarrow +infty}{lim}p_n(1/x)=a$.

If anyone had any hints for the problem and/or can point out if my thinking about this is on the right track, then I'd greatly appreciate it.

Hint: Since $fin mathcal{C}([1,infty))$ we have that $fcirc 1/xin mathcal{C}((0,1])$. Now, since $lim_{xtoinfty}f(x)=a$, $lim_{xto 0^+}fcirc 1/x=a$. What does this then tell us about the function
$$g:[0,1]tomathbb{R},qquad g(x)=begin{cases}
fcirc 1/x & xin (0,1] \
a & x=0
end{cases}?$$