Does such a polynomial always exist, for any pole of a rational function? Let $S \subseteq \mathbb{R}$ denote a cofinite subset of $\mathbb{R}$, and suppose $r : S \rightarrow \mathbb{R}$ is a rational function. Suppose $a$ is an element of $S^c$ (i.e. suppose $a$ is a pole of $r$.) Does there necessarily exist a polynomial function $p : \mathbb{R} \rightarrow \mathbb{R}$ such that the following limit exists and equals zero?
$$\lim_{x \rightarrow a}\left|r(x)-p\left(\frac{1}{x-a}\right)\right|=0$$
The idea is that $p\left(\frac{1}{x-a}\right)$ ought to work kind of like a vertical asymptote to $r(x)$ at $x=a,$ while providing more information than a mere vertical line about the behaviour of $r$ near this point.

Yes, Phrases to look up: Laurent series, principal part.
