Definition:Asymptotic Equality/General Definition/Point

Definition
Let $X$ be a topological space.

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

Let $f,g:X\to V$ be functions.

Let $x_0\in X$.

Then $f$ is asymptotically equal to $g$ as $x\to x_0$ iff $f-g=o(g)$ as $x\to x_0$.

Here $o$ denotes little-O notation.