Limit (Vector)

vector-calculus

Definition

Limit (Vector)

Let $D\subseteq {\mathbb{R}}^n$ and $f:D\to \mathbb{R}$. The limit ${\lim }_{x\to x_{0}f(\mathbf{x})}$ is a number $c$ with the following property:

$$\begin{array}{rl}\underset{\mathbf{x}\to {\mathbf{x}}_0}{\lim }& =c\\ & :⟺\\ & \forall \epsilon >0:\exists \delta >0:\forall \mathbf{x}\in D:0<||\mathbf{x}-{\mathbf{x}}_0|{|}_2<\delta ⟹|f(\mathbf{x})-c|<\epsilon \end{array}$$