Chapter 4 - Limits
Section 4.1 - Limits of Functions
Definition. Let \(A \subseteq \mathbb{R}\). Then, a point \(c \in \mathbb{R}\) is a cluster point of \(A\) if for every \(\delta > 0\), the \(\delta\)-neighborhood of \(c\) contains a point \(a \in A\) such that \(a \neq c\). That is, there exists some \(a\) such that \(0 < |a - c| < \delta\).
Theorem. A real number \(c\) is a cluster point for a set \(A\) if and only if there exists a sequence \((a_n)\) in \(A\\ \{c\}\) such that \(a_n \rightarrow c\)
Corollary. A real number \(c\) is a cluster point of a set \(A\) if and only if every \(\delta\)-neighborhood contains infinitely many points of \(A\).
Definition. The set of every cluster point of \(A\) is called the derived set of \(A\), and denoted \(A'\).
Corollary. A set \(A\) is closed if and only if \(A' \subseteq A\).
Remark. If \(A'\) is the derived set of \(A\), then \(A'' \subseteq A'\).
Remark. Intervals involving infinity and square brackets for the constant are closed.
Definition. Suppose \(f: A \rightarrow \mathbb{R}\) is a function with domain \(A \subseteq \mathbb{R}\), and let \(c \in A\) be a cluster point of \(A\). then, a real number \(L\) is a limit of \(f\) at \(c\) if given any \(\varepsilon > 0\), there exists some \(\delta > 0\) such that
Theorem. For a given function and cluster point, there can be at most one limit at said point.
Theorem. Let \(A \subseteq \mathbb{R}\) and \(f: A \rightarrow \mathbb{R}\). Then, to show that \(lim_{x \rightarrow c} f(x) = L\), it suffices to show that for every sequence \((a_n)\) in \(A\\ \{c\}\), the sequence \((f(a_n))\) converges tto \(L\).
Definition. The extended real numbers are \(\hat{\mathbb{R}} = \mathbb{R} \cup \{ \infty, -\infty \}\) are a totally-ordered set with supremum and infimum. Note that this set is no longer a field.
Definition. At any point \(c\), the limit of \(f\) at \(c\) is infinite if given some \(\alpha\), there exists some \(V_\delta(c)\) such that for all \(x \in V_\varepsilon(c)\), then \(f(x) \in V_\alpha(\infty)\).
Definition. The limit of a function at infinity is defined if for a given \(\varepsilon\), there exists some \(\alpha\) so that there exists some \(V_\delta(c)\) such that for all \(x \in A\),
Section 4.2 - Limit Theorems
Definition. Let \(A \subseteq \mathbb{R}\) and \(c \in \mathbb{R}\) be a cluster point of \(A\). Then, a function \(f: A \rightarrow \mathbb{R}\) is bounded on a neighborhood of \(c\) if there exists some \(\delta\)-neighborhood \(V_\delta(c)\) of \(c\) and some constant \(M > 0\) such that for all \(x \in A \cap V_\delta(c)\), then \(|f(x)| \leq M\).
Theorem. If \(A \subseteq \mathbb{R}\) and \(f: A \rightarrow \mathbb{R}\) has a finite limit at \(c \in \mathbb{R}\), then \(f\) is bounded on some neighborhood of \(c\).
Theorem. With \(A \subseteq \mathbb{R}\), and \(f, g: A \rightarrow \mathbb{R}\), with \(c \in \mathbb{R}\) a cluster point of \(A\), then if \(\lim_{x \rightarrow c} f(x) = L\) and \(\lim_{x \rightarrow c} g(x) = M\), then:
Additionally, if \(g(x) \neq 0\) for all \(x \in A\), and \(M \neq 0\), then
Corollary. If \(p, q \in \mathbb{R}[x]\), and \(q(c) \neq 0\) for some \(c \in \mathbb{R}\), then
Theorem. Squeeze Theorem. Let \(A \subseteq \mathbb{R}\). Then, if \(f, g, h: A \rightarrow \mathbb{R}\) and with \(c \in \mathbb{R}\) being a cluster point of \(A\), then if both
Then, \(\lim_{x \rightarrow c} g(x) = L\).