Definition:Big-O Notation/Real/Point

Definition
Let $x_0\in R$.

Let $f$ and $g$ be real functions defined on an open interval containing $x_0$.

The statement:
 * $f(x) = \mathcal O \left({g(x)}\right)$ as $x\to x_0$

is equivalent to the statement:
 * $\displaystyle \exists c\in \R: c\ge 0 : \exists \delta\in\R:\delta>0 : |f(x)|\leq c\cdot|g(x)|$ for all $x\in]x_0-\delta,x_0+\delta[$