Definition:Little-O Notation/Real

Definition

Estimate at infinity

Let $\map g x \ne 0$ for $x$ sufficiently large.

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

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

Also known as

The little-$\oo$ notation, along with big-$\OO$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.

Examples

Example: Sine Function at $+\infty$

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 $x \to \infty$.

Example: $x = \map \oo {x^2}$ at $+\infty$

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 $x \to \infty$.