In this answer it is explained how to numerically solve differential equations using (truncated) power series expansions. This can be useful within some interval of real line. Let us assume some localized truncated power series expansion:
$$P(x) = \sum_{k=0}^n c_k(x-x_0)^k$$
But as we know monomials grow so quickly that if $\|x-x_0\|\geq 1+\epsilon$, the function value increases very fast. In fact I think we can prove it will be $O(\epsilon^n)$.
Due to the power of compound interest (for you economics nerds) this grows so incredibly fast, so in practice we can only be sure about reasonably nice behaviour within $$\|x-x_0\|\leq 1-\epsilon$$ which will clearly be bounded by $1$ for each individual monomial term.
What techniques can we then use to make a workable approximation of some function anywhere outside of $\|x\|<1$?