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Limits for Advanced Students

Advith

2024 October 18

1 Introduction

  • 1.1. We have studied limitas for sequences, series, and functions - we will study limits for sequences of functions later on. However, our discussion has remained informal. We have not talked about what a limit means, how it is defined, and why the definitions are structured the way they are. This document will fill in those details.

  • 1.2. None of this material will be required for the course and it will not come up on the homework or exams. This is just to satiate those students who are curious about the mathematics.

2 Sequences

  • 2.1. Mathematics, both pure and applied, is filled with infinite processes. Examples of ininite series, algorithms, functions, tilings, flows, integrals, traces, and derivatives.

    It is very useful for us to be able to assign values to infinite processes. We want those values, in some manner, to reflect the end-result of those processes, and we would like those values to be assigned consistently. Our general framework for assigining these values will be called limits.

  • 2.2. It is most useful to start with sequences of numbers. We can then use limits of sequences to build limits for series and functions and more.

  • 2.3 Definition. We will use the symbol \(\mathbb N\) to refer to the natural numbers, \(\mathbb N = \{0,1,2,3,\dots \}\). The symbol \(\mathbb R\) will refer to the real numbers, which includes numbers like \(4.104\), \(\sqrt {2}\) and \(\pi \).

    A sequence is an ordered list of real numbers \(c_i\in \mathbb R\):

    \begin{equation} (c_0,c_1,c_2,c_3,c_4, \dots ) \end{equation}

    Sometimes it is useful to think about a sequence as a function \(c\), which will map the natural number \(n\) to the value \(c_n\) which we write as \(c: n \mapsto c_n\). This makes \(c\) a function from the natural numbers to the real numbers: \(c: \mathbb N \to \mathbb R\)

  • 2.4 Idea. We will not formally define what a limit of a sequence is. While there is a formal definition, it is a little technical at this level. The important point is that we say a sequence \(c_n \to L\) if we can make \(c_n\) as close as we want to \(L\) for large values of \(n\).