Definition:Little-O Notation/Real/Infinity/Definition 1

Definition
Let $f$ and $g$ be real functions defined on a neighborhood of $+ \infty$ in $\R$. Let $\map g x \ne 0$ for $x$ sufficiently large.

$f$ is little-$\oo$ of $g$ as $x \to \infty$ :
 * $\ds \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = 0$

Also see

 * Equivalence of Definitions of Little-O Notation for Real Functions at Infinity