What is the standard asymptotic notation when $x\rightarrow 0$

Question

I understand big "$\mathcal{O}$” and little "o" notation but every definition I have seen is for $n\rightarrow\infty$. But what about something like $f(x)\in o(g(x))$ for $x\rightarrow 0$? Is there a standard notation for this? Is it simply determined by context?

Answer 1

The o-notations are just shorthand for limiting processes, so we must still state near what point in the domain we're taking limits. It only isn't always included because it is usually clear from context.

But the particular one you mention is important, as it is an avenue to generalising single variable real calculus to multidimensional domains. It just means that the absolute ratio of $f$ and $g$ goes to $0$ as $x\to 0,$ or roughly speaking, that $f$ rushes to $0$ faster than $g.$