Definition of asymptotic relation

1. Jul 19, 2006

hanson

Hi all!

Can anyone explain to me why the asymptotic relation between a function and a power series is defined in such a way:

For all N,

$$f(x) - \sum_{n=0}^N a_n(x-x_0)^n << (x-x_0)^N$$

How does this incorporate the idea of asymptoticity?

$$\lim_{x\to{x}_{0}}\frac{f(x) - \sum_{n=0}^N a_n(x-x_0)^n}{(x-x_{0})^{N}}=0$$