Definition:Asymptotic Equality/General Definition/Point

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\, \cdot \,}$.

Let $f, g: S \to V$ be mappings.

Let $x_0 \in X$.

Then:

$f$ is asymptotically equal to $g$ as $x \to x_0$

if and only if:

$f - g = \map \oo g$ as $x \to x_0$

where $\oo$ denotes little-$\oo$ notation.