A key point about the Taylor series is that they are -infintesimal-
approximations.
Symbolically,
$f(x)-p_n(x) \in o(x^n)$
where $p_n$ is the $n$th Taylor approximation.
This generalizes:
- continuity (the $0$th order approx)
- differentiability (the $1$st order approx)
  ...i.e., the tangent line is the $1$st order approx,
  and the Taylor polynomials are higher order approx.
(in Physics, rarely go more than a few orders: $2$ or $3$, maybe up to $7$)

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note: the counterexamples:
$e^{-1/x^2}$ and the bump function
demonstrate -different- things
- the first that you can't go
  from germ/infintesimal info -> local info
  (just b/c all derivatives vanish doesn't mean it's locally constant)
- the second that you can't go
  from local info -> global
  (just b/c you vanish locally doesn't mean you vanish globally)
