Limits (Real Vector Functions)#

Definition: Limit (Real Vector Functions)

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function and let \(\boldsymbol{p}\) be a limit point of \(\mathcal{D}\).

We say that \(\boldsymbol{L} \in \mathbb{R}^n\) is the limit of \(f\) at \(\boldsymbol{p}\) if for each \(\varepsilon \gt 0\), there exists some \(\delta \gt 0\) such that for all \(\boldsymbol{x} \in \mathcal{D}\), we have the following:

\[0 \lt ||\boldsymbol{x} - \boldsymbol{p}||_{\mathbb{R}^m} \lt \delta \implies ||f(\boldsymbol{x}) - \boldsymbol{L}||_{\mathbb{R}^n} \lt \varepsilon\]

Notation

\[\lim_{x \to \boldsymbol{p}} f(\boldsymbol{x}) = \boldsymbol{L}\]

Theorem: Limit via Component Functions

The limit of a real vector function \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) at an accumulation point \(\boldsymbol{p}\) of \(\mathcal{D}\) is \(\boldsymbol{L} = \begin{bmatrix}L_1 & \cdots & L_n\end{bmatrix}^{\mathsf{T}} \in \mathbb{R}^n\) if and only if the limits of its component functions \(f_1, \dotsc, f_n\) are \(L_1, \dotsc, L_n\), respectively:

\[\lim_{\boldsymbol{x}\to \boldsymbol{p}} f(\boldsymbol{x}) = \boldsymbol{L} \iff \lim_{\boldsymbol{x}\to \boldsymbol{p}} f_k(\boldsymbol{x}) = L_k \qquad \forall k \in \{1,\dotsc,n\}\]

Proof

TODO

Theorem: Linearity of the Limit Operator

If \(f, g: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) have limits for \(\mathbf{x} \to \mathbf{p}\), then for all \(\lambda, \mu \in \mathbb{R}\):

\[\lim_{\mathbf{x} \to \mathbf{p}} [\lambda f(\mathbf{x}) + \mu g(\mathbf{x})] = \lambda \lim_{\mathbf{x} \to \mathbf{p}} f(\mathbf{x}) + \mu \lim_{\mathbf{x} \to \mathbf{p}} g(\mathbf{x})\]

Proof

TODO