What function is zero at zero, strictly positive elsewhere, infinitely differentiable at zero and has all...

exp(-1/x^2)

There are infinitely many other such functions.

This tells us why Taylor Series are a more limited device than they might be.
We form a Taylor series by looking at the derivatives of a function at a given
point; but this example shows us that the derivatives at a point may tell us
almost nothing about its behavior away from that point.