(Evil laugh) However in the 50's in Quantum Field Theory terms in the perturbation expansion were sometimes ignored because they were infinite...

I seem to recall a number of topological proofs (based on the series definition of continuity) that depend upon dropping small terms as well.

And aren't affine spaces based on linear approximations? I'm not sure about this one because I don't have much experience with affine spaces. Something about them being a "flat" approximation to a curved space.

-Dan