An underlying principle of the "C" quarter of calc is:
functions as limits

I can think of 3 ways of writing a function as a limit
worth studying/mentioning, at least:
- Taylor series
- Fourier series
- infinite product
  (these are a bit delicate: see Ahlfors)

Roughly:
Taylor = infinite polynomials, viewed as sum
  (coefficients and powers)
infinite products = infinite polynomials, viewed as product
  (zeros and multiplicities)
Fourier = infinite -trig- polys, viewed as sum

It's worth contrasting the following examples:
- Taylor series of $\sin x$
- product developement of $\sin x$ (or rather, $(\sin \pi x) / \pi$)
- Fourier series of $x$ (on $[-\pi,\pi]$):
    contrast approx sine by polys
    to approx polys by sine
