Everybody knows about Taylor approximations, right? These guys:

They can approximate function's behaviour near a certain point using polynomials and values of the function's derivatives near that same point. Despite their ubiquity, they are not the only way to approximate a function. Meet Pade approximant. They use rational functions, instead of polynomials.

Let's take a look at sinus for example:

You can see these fat tails that go anywhere but in the right direction. This is the destiny of any Taylor approximant, they don't get periodic behaviour very well.

Compare this to an Pade approximant of the order 6.

This approximant is kind of "aware" of infinities and tries to capture the local behaviour as best as it can.

You can play around with both approximants here.