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 $x_0=+\mathrm{\infty }$. Let $f:D\to ℝ$ 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

$$\underset{x\to x_0}{lim}\frac{g(x)}{f(x)}=0.$$

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”.