Question on definition of little o

Question

I would like to generalize the definition of little o.

The definition from Wikipedia is as such:

Let $f$ and $g$ be two real valued functions. We write $f(x) = o(g(x))$ as $x \to \infty$ if for all $\epsilon > 0$, there exists $x_0$ such that $|f(x)| \leq \epsilon |g(x)|$ whenever $x \geq x_0$.

I would like to extend this definition so that $x \to a$ where $a$ is some real number, and not necessarily infinity. How does this change the definition?

Answer 1

Replace the ending “there exists …” with

… there exists $\delta>0$ such that $|f(x)|\le\epsilon|g(x)|$ whenever $0<|x-a|<\delta$.