A Concept of Limits

By Donald W. Hight

An exploration of conceptual foundations and the practical applications of limits in mathematics, this text offers a concise introduction to the theoretical study of calculus. It analyzes the idea of a generalized limit and explains sequences and functions to those for whom intuition cannot suffice. Many exercises with solutions. 1966 edition.

• Language: English
• Publisher: Dover Publications
• Release date: Jul 17, 2012
• ISBN: 9780486153124

Book preview

A CONCEPT

OF LIMITS

Donald W. Hight

Department of Mathematics

Pittsburg State University

Dover Publications, Inc., New York

To My Parents—

My Appreciation of Them Is

tone Increasing and Unbounded Above

Copyright © 1966, 1977 by Donald W. Hight.

All rights reserved.

   This Dover edition, first published in 1977, is a corrected republication of the work first published by Prentice-Hall, Inc., Englewood Cliffs, N. J., in 1966. The section entitled Answers to Even-Numbered Exercises, published in 1966 by Prentice-Hall as a separate supplement, is here republished as an integral part of the book.

International Standard Book Number: eISBN 13: 978-0-486-15312-4

Library of Congress Catalog Card Number: 77-80029

Manufactured in the United States by Courier Corporation

63543006

www.doverpublications.com

Contents

1SEQUENCES AND THEIR LIMITS

1–1 Infinite Sequences

1–2 Graphs of Infinite Sequences

1–3 Some Examples to Ponder

1–4 Types of Sequences

1–5 Circumference of a Circle

1–6 Sequences in the Physical World and in History

1–7 The Limit of a Sequence: Informal

1–8 A Precise Language

1–9 Limit of a Sequence

1–10 Graphical Interpretation of the Limit

1–11 Theorems on Limits

1–12 Applications of Limits of Sequences

2FUNCTIONS AND THEIR LIMITS

2–1 Functions

2–2 Graphs of Functions

2–3 Some Examples to Ponder

2–4 Classification of Functions

2–5 A Limit of a Function: Informal

2–6 The Limit At-The-Right

2–7 The Limit At-The-Left

2–8 Theorems on Limits

2–9 Another Type of Limit

2–10 More Examples to Ponder

2–11 Limits at a Real Number b

2–12 More Theorems on Limits

2–13 Some Special Limits

2–14 Continuity

3GENERALIZATION AND APPLICATION OF THE LIMIT CONCEPT

3–1 Arithmetic of Functions

3–2 Interrelation of Limit Definitions

3–3 The Generalized Limit

3–4 Generalized Limit Theorems

3–5 More on Composition and Continuity

3–6 Limits in High School Mathematics

3–7 On from Here…

3–8 Overview

GLOSSARY OF SYMBOLS

BIBLIOGRAPHY

ANSWERS TO ODD-NUMBERED EXERCISES

ANSWERS TO EVEN-NUMBERED EXERCISES

GRAPHS OF EXAMPLES TO PONDER

INDEX

Preface

LIMITS HAVE occupied a unique position in mathematics education. In secondary school or precalculus mathematics, they have been regarded as too hard and thus avoided; in calculus courses, they have been taken for granted or skimmed over hurriedly as if they were intuitively obvious. The purpose of this book is to extend a concept of limits from intuitive ideas about limits to knowledge of a generalized limit that is applicable in many areas of mathematics. To achieve this goal, a method has been employed that provides understanding through participation. Considerable effort has been made to allow and encourage each reader to progress at his own rate through a sequence of developmental steps from a known concept to a desired conclusion. Therefore, the book begins very slowly and assumes knowledge of high school algebra and an acquaintance with trigonometry. Then, it progresses through historical accounts of limits, the limits of sequences and functions, continuity, and proofs of theorems to the development and applications of a unifying concept of a generalized limit.

In addition to an understanding of a concept of limits, the book offers the following: motivated review and reinforcement of familiar topics such as algebra, inequalities, absolute value, functions, graphs, trigonometry, series, and geometry related to the measure of a circle; a thorough mathematical study of an important concept; the appreciation of definitions, proofs, and a mathematical structure; transition from typical precalculus and traditional mathematics to more sophisticated contemporary analysis; and some experience at limit-guessing and "δ-finding."

Chapter 1 is devoted to an intuitive and historical development of the limit of a sequence. The definition of a limit is preceded by Examples to Ponder, geometrical and historical examples, graphs, and repeated questions such as, "For what values of n is sn within ε of L?" Experience with the concept is provided by some simple theorems, proofs that some real number L is or is not the limit of a sequence 5, and helpful applications.

Chapter 2 presents standard limits of functions and continuity. Although a sequence was called a function in Chapter 1, functions are presented in greater detail in this section. Stress is placed upon simple theorems concerned with boundedness, uniqueness of limits, and positive or negative limits. Examples and exercises emphasize the interrelation of various types of limits, special limits, and continuity of trigonometric functions.

In Chapter 3, unifying attributes of the defined limits of sequences and functions are organized into a generalized limit. Then, by proving a single theorem for the generalized limit, a proof is given that is readily applicable to every type of limit previously discussed. The chapter also introduces limits and continuity of sums, differences, products, quotients, and composites of functions and concludes with applications of limits to high school mathematics, to calculus, and to extensions of the generalized limit theorems.

Mathematical terms that are especially associated with the presentation are written in boldface when they are introduced; similarly, important terms that have more widespread usage are written in italics.

and "lim {(n, sn)} with lim sn are used to stress that a limit of a function or sequence has been defined and not a limit of a set of range values. Since students today may not think of a variable as a quantity that changes, reference to active variables such as x increases without bounds, f(x) approaches 1, x → ∞, and f(x) → 1"are avoided. Instead, sets of numbers and their relationship are used to present the ε, δ concept throughout, from intuitive examples to definitions. Furthermore, such names as "limit at b and limit at-the-right are used for limit as x approaches b and limit as x tends toward infinity. Another innovation, graphs of sequences on n-inverted" coordinate systems, has proved to be quite successful in practice. Also, the presentation of a generalized limit not only allows opportunity to unify the concept, but the proof of one generalized limit theorem provides proofs of theorems for six types of familiar limits plus other limits that readers may subsequently encounter.

I wish to express my gratitude and indebtedness to Dr. Bruce E. Meserve for his constructive criticism, guidance, and encouragement during the preparation of the book. Also, thanks are due to Dr. R. G. Smith, Dr. Glen Haddock, to my wife Betty, and to the many others who assisted me in preparing and testing this material.

Donald W. Hight

chapter 1

Sequences and Their Limits

This book is intended to be read with a pencil in hand. It is not designed .to be read as a story, for unless you knew in advance about limits you would soon be confused and lost. Many examples are given which you should analyze and classify. Questions are asked which you are to ponder and answer for yourself. It will then be possible (let us hope) for you to anticipate subsequent considerations and eventually to grasp for yourself a limit concept and to obtain for yourself acceptable definitions of limits. Now, if you have not already done so, get a pencil in hand and a pad of paper beside your book so that we may start our explorations.

1–1 Infinite Sequences

You already have some idea of what a sequence is, for the word is common. In referring to a sequence of events you want to communicate that one event happened, then the next, and the next, and so forth. We wish to define an infinite numerical sequence in a similar but more specific manner. To specify that there is a first event and then a next event and a next, and so forth, we utilize the natural numbers (or the set of positive integers). The events that we consider are real numbers and are called terms of the sequence. An infinite sequence of real numbers is a function in which each natural number is associated with a unique real number. Since we are concerned in this text only with infinite sequences of real numbers, we shall refer to them simply as sequences.

We shall express a sequence in a traditional manner as an ordered set or list and also as a set of ordered pairs. Thus a sequence s may be expressed either as a list or ordered set,

or as a set of ordered pairs of related numbers,

Here as throughout this book, we shall use the letter "n" as a symbol for a natural number. Thus, each expression for the sequence indicates that 1 is associated with s1, 2 is associated with s2, 3 is associated with s3, and, in general, every natural number n is associated with a unique real number sn.

Example 1 Express (a) as a list and (b) as a set of ordered pairs the sequence in which sn = 2n.

Not all sequences can conveniently be expressed by a simple algebraic equation involving n. Some are better suited to a general expression accompanied by an explanation.

Example 2 Express (a) as a list and (b) + 1) and each even natural number e is associated with 2(e – 1).

The domain of a sequence (infinite sequence) is the set of natural numbers. The range of a sequence is the set of terms of the sequence. We should recall the accepted usage of braces to express a set and be reminded that an expression of a set does not indicate any ordering of its elements. Thus the domain of every sequence is generally expressed as {1, 2, 3, 4,…, n, …}, where the natural numbers are listed in their natural order by habit and for convenience. We are denoting the set of natural numbers and nothing else. The ranges of the sequences in Examples 1 and 2 are the same set, the set of even natural numbers. This set could be expressed as {2, 4, 6, 8, … , 2n, …} or simply as S if we defined S to be the set of even natural numbers.

Two sequences s and t are equal if and only if sn = tn for every natural number n. The sequences in Examples 1 and 2 are sequences that are not equal. However, their domains are equal (the same set—the set of natural numbers) and their ranges are equal (the same set—the set of even natural numbers).

Example 3 (a) List the first five terms of the sequence {(n, sn)} in which sn = ( – 1)n. (b) Give the range of the sequence.

Exercises

1. List the first five terms of the sequence s = {(n, sn)} in which sn = 2n – 1.

2. List the first five terms of the sequence t = {(n, tn)} in which tn = 1/n.

3. Consider the sequence c = 2, 2, 2, 2, … , 2, … in which cn = 2 for each n. (a) State the range of the sequence c. (b) Represent the sequence in the form {(n, sn)}.

4. Give a representation in the form {(n, sn)} of a sequence whose range is {1}. (This sequence and the sequence in Exercise 3 are called constant sequences because they may be represented in the form {(n, c)} for some constant c = cn for every natural number n.)

5. Consider the sequence 1, 2, 3, 3, … , 3, … in which s1 = 1, s2 = 2, and sn = 3 for each natural number n ≥ 3. Name two other sequences that have the same range.

6. Let the first term of a sequence {(n, an)} be some given real number, say c, and let an = c + (n – 1)d, where d is a given real number. Write the first five terms of this sequence. (Such a sequence is called an arithmetic sequence and can be expressed by "{(n, c + (n – 1)d)}.")

7. Let the first term of a sequence g be some given real number, say a where a ≠ 0, and let gn = arn – ¹, where r is a given real number. Write the first five terms of the sequence g. (Such a sequence is called a geometric sequence and is expressed by "{(n, arn – ¹)}")

8. Let the first two terms of a sequence s each be 1 and sn = sn – 1 + sn – 2 for each natural number n ≥ 3. For example, s3 = 1 + 1 = 2. Write the first seven terms of this sequence. (This sequence is called a Fibonacci sequence.)

1–2 Graphs of Infinite Sequences

To stimulate your intuition and sharpen your perception let us consider some graphs of sequences. The graph of a sequence s is the set of all points whose coordinates are (n, sn) on a coordinate system with an n-axis and an sn-axis. As you make graphs and look at them, do not hestitate to look for new truths or relationships. However, do not be completely satisfied that the impressions of your eye are valid until a proof is given using previously accepted theorems, definitions, or properties of numbers.

Example 1 .

Example 2 Graph on a Cartesian coordinate system the sequence {(n, 1)}.

The sequences graphed in Examples 1 and 2 are infinite sequences and their graphs are infinite sets of points. The graphs extend indefinitely off the paper because our sequences are infinite sequences, and because we associate with each n the point whose distance is n units to the right of 0. One way that this difficulty can be overcome for infinite sequences is by associating with each natural number n the point whose distance is 1/n units to the left of 0. In this scheme the graph of the number pair (n, sn) will be a point at a distance 1/n units to the left of 0 and at a distance | sn | up or down fromO. Henceforth in this book such a graph will be referred to as a graph on an n-inverted coordinate system.

Example 3 Graph on an n-inverted