Definition of Limit [= ($\epsilon$-$\delta$) definition] Assume that $S\subseteq \mathbb{R}^n$, and that $f:S\to \mathbb{R}$ is a multivariate function. The statement$$ \underset{\textbf{x} \to \textbf{a}}{\lim} f (\textbf{x})= L$$is defined to mean that$$\begin{equation}\label{lim.def}\forall \varepsilon >0, \ \ \exists \delta>0 \quad\mbox{ such that if } \mathbf x \in S \mbox{ and } 0 \end{equ..