Little o notation

Definition

Let $f$ and $g$ be real functions defined on an interval $[x_0, \infty)$. The notation $f(x) = o(g(x))$ as $x \to \infty$ means that for each $\varepsilon > 0$ there is $x_1 \ge x_0$ with $$|f(x)| \le \varepsilon |g(x)| \quad \text{for all } x \ge x_1.$$

Remarks

If $g(x) \ne 0$ for large $x$, then $f = o(g)$ means that $f(x)/g(x) \to 0$. Each $f = o(g)$ is also $f = O(g)$, in the sense of big O notation. For example, $x = o(x^2)$ and $\ln x = o(x)$ as $x \to \infty$. Mathlib defines it for filters, as Asymptotics.IsLittleO.

Explanations

L1 human

The statement $f = o(g)$ says that $f$ becomes negligible compared with $g$. Each fixed fraction of $|g|$ eventually exceeds $|f|$.