Rigorous mathematical definition of "much greater than" symbol

Question

What does $f(x) \gg g(x)$ mean mathematically? How can we characterize "much greater than"?

Answer 1

It depends on the context and what kind of functions $f$ and $g$ are, etc. One way is to say that $f(x) \gg g(x)$ if for any constant $c>0$ there exists some value $x_0 \in \mathbb{R}$ such that $f(x) > c g(x)$ for all $x > x_0$.

Answer 2

I usually just think like this

$$f(x) + g(x) \sim f(x)$$

That is $g(x)$ can be neglected