Definition:Little-O Notation/Real/Infinity
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Definition
Let $f$ and $g$ be real functions defined on a neighborhood of $+ \infty$ in $\R$.
Definition 1
Let $\map g x \ne 0$ for $x$ sufficiently large.
$f$ is little-$\oo$ of $g$ as $x \to \infty$ if and only if:
- $\ds \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = 0$
Definition 2
$f$ 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-$\oo$ $g$.
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 $x \to \infty$.