Definition:Little-O Notation/Real/Infinity

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 $x \to \infty$ if and only if:

$\ds \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = 0$

$f$ is little-$\oo$ of $g$ as $x \to \infty$ if and only if:

$\forall \epsilon \in \R_{> 0}: \exists x_0 \in \R: \forall x \in \R: x \ge x_0 \implies \cmod {\map f x} \le \epsilon \cdot \cmod {\map g x}$

This is denoted:

$f = \map o g \qquad \paren {x \to \infty}$

This statement is voiced $f$ is little-$\oo$ of $g$ or simply $f$ is little-$\oo$ $g$.

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \sin x$

Then:

$\map f x = \map \oo x$

as $x \to \infty$.

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = x$

Then:

$\map f x = \map \oo {x^2}$

as $x \to \infty$.