limit
Table of Contents
1. Introduction
A limit in mathematics is a tool used to describe the intuitive process of a value or a set of values tending towards another. First, we will define limits as they pertain to sequences, and then we will define them on functions. For a sequence \(\{s_{n}\}\):
\begin{align*} \lim s_{n} = s \iff \forall \epsilon > 0, \exists N , n > N \implies | s_{n} - s | < \epsilon \end{align*}What this means is that at some point in the sequence, for some choice of epsilon, no matter how small it is, there has to be an index where every term after that index is closer to \(s\) than epsilon. If some single number \(s\) and sequence \(\{s_{n}\}\) fulfills this criteria, then it is said that the limit of the sequence is \(s\). Generally speaking, we use the set \(\mathbb{R} \cup \{ -\infty, +\infty \}\), where there is a natural ordering:
\begin{align*} \forall a \in \mathbb{R}, - \infty < a < +\infty \end{align*}defined. Note that we can define equivalence relations on these symbols, but algebra reamins undefined.
1.1. Unbounded Sequences
Unbounded sequences can still limit to \(+\infty\) or \(-\infty\), although the limit does not exist for many unbounded sequences. If a sequence is one of:
- unbounded above
- unbounded below
but not both, it is possible that such sequences limit to \(\infty\).
1.2. Limits on Monotone Sequences
An increasing sequence is a sequence \(\{s_{n}\}\) defined such that:
\begin{align*} \forall n \in \mathbb{N}, \forall m \in \mathbb{N}, n \ge m \implies s_{n} \ge s_{m}. \end{align*}The limit of monotone sequences always exists.
We know:
\begin{align*} \lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies | s_{n} - s | < \epsilon \\ \end{align*}which is equivalent to:
\begin{align*} \lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies s - \epsilon < s_{n} < s + \epsilon \end{align*}and our sequence \(\{s_{n}\}\) is monotone. If \(\{s_{n}\}\) is increasing, we have:
\begin{align*} s_{n + 1} \ge s_{n} \end{align*}for all n. Without loss of generality we shall assume \(\{s_{n}\}\) is increasing. Then we take two cases:
- \(\{s_{n}\}\) is bounded.
- \(\{s_{n}\}\) is unbounded.
In the case \(\{s_{n}\}\) is bounded:
\begin{align} \label{} \exists M, \forall n, s_{n} \le M \\ s_{0} \le ... \le s_{n} \le s_{n + 1} \le s_{n + 2} \le ... \le M \end{align}