User:IssaRice/Little o notation

Definition

Definition (little o near a point). Let $f:\mathbf {R} \to \mathbf {R}$ and $g:\mathbf {R} \to \mathbf {R}$ be two functions, and let $a\in \mathbf {R}$. We say that $f$ is little o of $g$ near $a$ iff for every $\epsilon >0$ there exists $\delta >0$ such that $|x-a|<\delta$ implies $|f(x)|<\epsilon |g(x)|$.

Definition (little o at infinity). Let $f:\mathbf {R} \to \mathbf {R}$ and $g:\mathbf {R} \to \mathbf {R}$ be two functions. We say that $f$ is little o of $g$ at infinity iff for every $\epsilon >0$ there exists $M$ such that for all $x$, $x>M$ implies $|f(x)|<\epsilon |g(x)|$.

Can we write just $f\in o(g)$ or $f=o(g)$ or $f(x)\in o(g(x))$ or $f(x)=o(g(x))$? In general we can't because for this notation to make sense, we also need to know where the argument $x$ is going. In algorithms, we have $x\to \infty$, but in analysis (e.g. in some definitions of differentiability) we have $x\to 0$