Exercises of Limits E-Book

Simone Malacrida

• Herausgeber: BookRix
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2023

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics: infinitesimals and infinities limits and forms of indeterminacy continuous functions and points of discontinuity Initial theoretical hints are also presented to make the performance of the exercises understandable.

Leseprobe

Simone Malacrida

Exercises of Limits

BookRix GmbH & Co. KG81371 Munich

Table of Contents

Table of Contents

“Exercises of Limits”

INTRODUCTION

LIMITS

CONTINUOUS FUNCTIONS

“Exercises of Limits”

“Exercises of Limits”

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

infinitesimals and infinities

limits and forms of indeterminacy

continuous functions and points of discontinuity

Initial theoretical hints are also presented to make the performance of the exercises understandable.

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

ANALYTICAL INDEX

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INTRODUCTION

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I – LIMITS

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

Exercise 10

Exercise 11

Exercise 12

Exercise 13

Exercise 14

Exercise 15

Exercise 16

Exercise 17

Exercise 18

Exercise 19

Exercise 20

Exercise 21

Exercise 22

Exercise 23

Exercise 24

Exercise 25

Exercise 26

Exercise 27

Exercise 28

Exercise 29

Exercise 30

Exercise 31

Exercise 32

Exercise 33

Exercise 34

Exercise 35

Exercise 36

Exercise 37

Exercise 38

Exercise 39

II – CONTINUOUS FUNCTIONS

Exercise 1

Exercise 2

Exercise 3

Exercise 4

INTRODUCTION

INTRODUCTION

In this exercise book, some examples of calculation relating to limits, forms of indeterminacy, the comparison between infinitesimals and the concept of continuity of a function are carried out.

Furthermore, the main theorems used in this area are presented.

The calculus of limits is the first result of mathematical analysis, without which it is impossible to construct and understand the subsequent differential and integral calculus.

In order to understand in more detail what is explained in the resolution of the exercises, the reference theoretical context is recalled at the beginning of each chapter.

What is presented in this workbook is generally addressed during the last year of scientific high schools and, more rigorously, in the first course of mathematical analysis at university level.

I

LIMITS

LIMITS

Given a function defined on a subset X of the set of real numbers and a point of accumulation of this subset, we define the limit of the function as x tending to the point of accumulation a real number such that the distance between it and the value of the function at the point is an infinitesimal. In formulas:

In this case we say that the limit as x tending to the point of accumulation of the function is given by l.

Equivalently we can say that for every neighborhood of l there exists a neighborhood of the accumulation point such that the function belongs to the neighborhood of l.

We point out that the point of accumulation is not necessarily contained in the domain of the function, i.e. the local vision is totally independent of the punctual one.

We can extend the concept of limit if the real number l is infinite. In this case it is valid:

The limit is written like this: