$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$
Compare behaviour of two functions $f(\mathbf{x})$ and $g(\mathbf{x})$ as $\mathbf{x}$ "tends to something" (in the usage it is clear).
Definition 1
Observe that (2) implies (1) but (1) does not imply (2) and (3) implies (4) but (4) does not imply (3).
See in details Wikipedia; also $\Omega$ notation (which we do not use).