2.3 KiB
Chapter 4 - Limits
Secrion 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 thhat 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
Collary. A real number c is a cluster point of a set A if and only if every $\delta$-neighborhood conttains infinitely many points of A.
Definition. The set of every cluster point of A is called the derived set of A, and denoted A'.
Collary. 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 goven any \epsilon > 0, there exists some \delta > 0 such that
0 < |x-c| < \delta \Rightarrrow |f(x) - L| < \epsilon
Therorem. 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 aree \hat{\mathbb{R}} = \mathbb{R} \cup \{ \inftyy, -\infty \} are a totally-ordered set witth supremum and infimum. Note that this set is no longer a field.
Definition. At any point c, the limitt of f at c is infinite if given some \alpha, there exists some V_\delta(c) such that forr all x \in V_\epsilon(c), then f(x) \in V_\alpha(\infty).
Definition. The limit of a function at infinity is defined if for a given \epsilon, there ixists some \alpha so that there exists some V_\delta(c) such that for all x \in A,
x > \alpha \Rightarrow |f(x) - L| < \epsilon