OK, here is a directly relevant answer: asymptotic expansions, which are finite expansions by definition, are used to approximate $f(x_{0}+h)$ and can be used with divergent series (That is what Poincaré invented them for) as well as convergent series.
The second part of my answer is that, at least in calculus, but in fact in precalculus, polynomial expansions (certainly finite and in fact, rather short) should be used to describe the qualitative behavior near $x_{0}$ and $\infty$ of the usual functions.
For a simple example, consider the function $f(x)=\frac{x^{3}-8}{x^{2}-5x+6}$.
- Near $\infty$, we get by long division in descending powers that $f(x) = x +5 + 19x^{-1} + o[x^{-1}]$ i.e. an oblique asymptote.
- Near $+2$, one of two possible poles, we get by long division in ascending powers that $f(+2+h) = \frac{+12h +6h^{2} +h^{3}}{-h+h^{2}} = -12 -18h -19h^{2} + o[h^{2}]$ i.e. that $+2$ is regular.
- Near $+3$, the other possible pole, we get by short division that $f(+3+h) = \frac{+19 +27h +9h^{2} +h^{3}}{h+h^{2}} = \frac{+19+ o[1]}{h+ o[h]}=+19h^{-1} + o[h^{-1}]$ i.e. that $+3$ is indeed a pole.
- A smooth interpolation of the local graphs near $\infty$ and $+3$---the local graph near $+2$ provides only confirmation---gives the essential global graph which in turn shows that $f$ has at least one maximum and one minimum.

In precalculus, rather than little ohs, I just use [...] to stand for "something too small to matter here."
See, for instance, my implementation of the above: Reasonable Algebraic Functions