[Understanding Analysis by Stephen Abbott] - [Chapter 4 Functional Limits and Continuity] - [4.2 Functional Limits]

Definition 4.2.1 (Functional Limit). Let \(f : A \to \mathbb{R}\), and let \(c\) be a limit point of domain \(A\). We say that \(\lim_{x \to c} f(x) = L\) provided that, for all \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\) (and \(x \in A\)) it follows that \(|f(x) - L < \epsilon|\)

Definition 4.2.1B (Functional Limit: Topological Version). Let \(c\) be a limit point of the domain of \(f : A \to \mathbb{R}\). We say \(\lim_{x \to c} f(x) = L\) provided that, for every \(\epsilon\)-neighborhood \(V_\epsilon (L)\) L, there exists a \(\delta\)-neighborhood \(V_\delta (c)\) around \(c\) with the property that for all \(x \in V_\delta (c)\) different from \(c\) (with \(x \in A\)) it follows that \(f(x) \in V_\delta (L)\)

Theorem 4.2.3 (Sequential Criterion for Functional Limits). Given a function \(f : A \to \mathbb{R}\) and a limit point \(c\) of \(A\), the following two statements are equivalent:

1. \(\lim\limits_{x \to c}f(x) = L\)
2. For all sequences \((x_n) \subseteq A\) satisfying \(x_n \neq c\) and \((x_n) \to c\), it follows that \(f(x_n) \to L\)

Corollary 4.2.4 (Algebraic Limit Theorem for Functional Limits). Let \(f\) and \(g\) be functions defined on a domain \(A \subseteq \mathbb{R}\), and assume \(\lim_{x \to c}f(x) = L\) and \(\lim_{x \to c}g(x) = M\) for some limit point \(c\) of \(A\). Then,

1. \(\lim\limits_{x \to c}kf(x) = kL\) for all \(k \in \mathbb{R}\)
2. \(\lim\limits_{x \to c}[f(x) + g(x)] = L + M\)
3. \(\lim\limits_{x \to c}[f(x)g(x)] = LM\)
4. \(\lim\limits_{x \to c}f(x)/g(x) = L/M\)

Corollary 4.2.5 (Divergence Criterion for Functional Limits). Let \(f\) be a function defined on \(A\), and let \(c\) be a limit point of \(A\). If there exists two sequence \((x_n)\) and \((y_n)\) in \(A\) with \(x_n \neq c\) and \(y_n \neq c\) and $$ \lim x_n = \lim y_n = c \text{ but } \lim f(x_n) \neq \lim f(y_n) $$ then we can conclude that the functional limit \(\lim_{x \to c}f(x)\) does not exist.