Two functions $f$, $g: \mathbb R \to \mathbb R$ are said to be equal up to order $n$ at $a$ if and only if $\lim_{x \to a} \frac{f(x) - g(x)}{(x - a)^n} = 0.$ What is the intuition behind this definition? What is the idea behind considering two functions to be "very equal" versus "not so equal"?
The current picture in my head is this: two functions are "somewhat equal" if their graphs "look like they overlap." Given $a \in \mathbb R$, two functions are equal up to a "high order" if you have to zoom in "a lot" at $a$ in order to tell the two graphs apart; they are equal up to a "low order" if you can see two distinct graphs even without zooming in at $a$? I don't really know how I got this picture; I think it's somewhat correct, but I can't see how the picture relates to the limit definition.