formal definition of Landau notation

Let us consider a domain $D$ and an accumulation point $x_0\in \overline D$ . Important examples are $D=\R$ and $x_0\in D$ or $D=\mathbb N$ and $x_0=+\infty$ . Let $f\colon D\to \R$ be any function. We are going to define the spaces $o(f)$ and $O(f)$ which are families of real functions defined on $D$ and which depend on the point $x_0\in \overline D$ .

Suppose first that there exists a neighbourhood $U$ of $x_0$ such that $f$ restricted to $U\cap D$ is always different from zero. We say that $g\in o(f)$ as $x\to x_0$ if$$ \lim_{x\to x_0} \frac{g(x)}{f(x)}=0.$$ We say that $g \in O(f)$ as $x\to x_0$ if there exists a neighbourhood $U$ of $x_0$ such that$$ \frac{g(x)}{f(x)} \text{is bounded if restricted to $D\cap U$}.$$ In the case when $f\equiv 0$ in a neighbourhood of $x_0$ , we define $o(f)=O(f)$ as the set of all functions $g$ which are null in a neighbourhood of $0$ .

The families $o$ and $O$ are usually called "small-o" and "big-o" or, sometimes, "small ordo", "big ordo".