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:
A further extension is given by the condition in which the point of accumulation is at infinity, ie that the upper bound of the set X is infinite. In that case:
Which leads to this definition of limit:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 22.04.2023
ISBN: 978-3-7554-3988-2
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