Definition:Asymptotically Equal/General Definition/Point

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

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

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

Let $x_0 \in X$.

Then:
 * $f$ is asymptotically equal to $g$ as $x \to x_0$


 * $f - g = o \left({g}\right)$ as $x \to x_0$
 * $f - g = o \left({g}\right)$ as $x \to x_0$

where $o$ denotes little-O notation.