Limits (Real Scalar Fields)#
In the case of real scalar fields, the definition of a limit reduces to the following.
Definition: Limit of a Real Scalar Field
Let \(f: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}\) be a real scalar field and let \(\mathbf{a} \in \mathbb{R}^n\) be an accumulation point of \(\mathcal{D}\).
A numbers \(L \in \mathbb{R}\) is the limit of \(f\) for \(\mathbf{x} \to \mathbf{a}\) if and only if for each \(\varepsilon \gt 0\) there exists some open ball \(B_{\delta}(\mathbf{a})\) around \(\mathbf{a}\) such that for all \(\mathbf{x} \in \mathcal{D}\) different from \(\mathbf{a}\),
Notation
Theorem: Heine Definition of Convergence
Let \(f: \mathcal{D}_f \subseteq \mathbb{R}^n \to \mathbb{R}\) be a real scalar field and let \(\mathbf{p} \in \mathbb{R}^n\) be an accumulation point of \(\mathcal{D}_f\).
The limit of \(f\) at \(\mathbf{p}\) is \(L \in \mathbb{R}\) if and only if for each real vector sequence \((\mathbf{x}_k)_{k \in \mathcal{I}}\) in \(\mathcal{D}_f \setminus \{\mathbf{p}\}\) which converges to \(\mathbf{p}\), the sequence \((f(\mathbf{x}_k))_{k \in \mathcal{I}}\) converges to \(L\).
Proof
TODO
Theorem: Algebraic Properties
Let \(f, g: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}\) be real scalar fields.
If the limits of \(f\) and \(g\) for \(\mathbf{x} \to \mathbf{a}\) exist, then
Furthermore, if \(g(\mathbf{x}) \ne 0\) for all \(\mathbf{x} \in \mathcal{D}\) and \(\displaystyle \lim_{\mathbf{x} \to \mathbf{a}} g(\mathbf{x}) \ne 0\), then
Proof
TODO