Definitions On Landau Notation (Big O and little o)

Question

What are the definitions on Big O and little o for when $x \in R^m$ approaches $s\in R^m$ And not $x$ going to infinity?

Answer 1

Definition:

$f(x) = \mathcal{O}({g(x)}) $ when $x\to x_0$

if and only if we can find $C>0$ and $\delta>0$ such that $|f(x)| \le C|g(x)|$ for $|x - x_0| < \delta$.

If $g(x)$ is non-zero,

$f(x) = \mathcal{o}({g(x)}) $ when $x\to x_0$ if and only if

$\lim \limits_{x\to x_0} {\frac{f(x)}{g(x)}} = 0$