DIFFERENTIAL CALCULUS: A Mathematical Analysis for Applied Sciences

By Alix Aurora Fuentes Medina

The textbook "DIFFERENTIAL CALCULUS. A Mathematical Analysis for Applied Sciences" is dedicated to all university students in distance or face-to-face programs who need to learn Differential Calculus to use its contents in Applied Sciences.

The relevant contributions of this textbook focus on the Pedagogy for the Development of Autonomous Learning together with the logical and complete selection of topics. These topics offer the student the necessary knowledge to achieve the conceptual and applied handling of Differential Calculus with mastery in applied sciences, without including demonstrative processes or theoretical constructions typical of pure mathematics.

The methodological design offers the student alternatives to learn and learn autonomously in a logical-analytical-constructive process and application of concepts.

The textbook "DIFFERENTIAL CALCULUS. A mathematical analysis for applied sciences" aims for university students to learn by themselves the fundamental theory of differential calculus as part of a teaching-learning process. This process mainly includes theoretical management and conceptual application oriented to socioeconomic and administrative sciences.

• Language: English
• Publisher: Lulu.com
• Release date: Jul 19, 2018
• ISBN: 9781387958122

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ILLUSTRATIONS

ILLUSTRATION 1. SYMBOLOGY OF THE LIMIT OF A FUNCTION…………….…

ILLUSTRATION 2. GRAPHICAL INTERPRETATION OF THE LATERAL LIMITS AND THE LIMIT OF A FUNCTION IN A POINT………………………………………………

ILLUSTRATION 3. ONE GRAPHIC INTERPRETATION OF THE INFINITE LIMIT…

ILLUSTRATION 4. GRAPHIC INTERPRETATION OF THE LIMIT IN THE INFINITE……………………………………………………………………………….…

ILLUSTRATION 5. THEORY OF THE EXISTENCE OF MAXIMUM AND MINIMUM VALUES…………………………………………………………………………..………..

ILLUSTRATION 6. INTERMEDIATE VALUE THEOREM………………………….…

ILLUSTRATION 7. AVERAGE RATE OF CHANGE OF A FUNCTION……….…..…

ILLUSTRATION 8. RATE OF THE AVERAGE VELOCITY…………………………..

ILLUSTRATION 9. STRAIGHT LINE SECANT TO THE CURVE………………..…

ILLUSTRATION 10. TANGENT STRAIGHT LINE TO THE CURVE IN A POINT….

ILLUSTRATION 11. DERIVATIVE OF A FUNCTION AT A POINT………………….

ILLUSTRATION 12. ALTERNATIVE FORM OF THE DERIVATIVE OF A FUNCTION AT A POINT………………………………………………………………………….…..

ILLUSTRATION 13. DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTION Sin-1x…………………………………………………………..…………………………..

ILLUSTRATION 14. DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTION Cos-1x………………………………………………………………………………………

ILLUSTRATION 15. DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTION Tan-1x…………………………………………………………………………………….

ILLUSTRATION 16. DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTION Cot-1x……………………………………………………………………………………..

ILLUSTRATION 17. DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTION Sec-1x………………………………………………………………………………………

ILLUSTRATION 18. DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTION Csc-1x……………………………………………………………………………………..

ILLUSTRATION 19. INCREASING FUNCTION………………………………………

ILLUSTRATION 20. DECREASING FUNCTION…………………………………..…

ILLUSTRATION 21. INNER EXTREME THEOREM…………………………………

ILLUSTRATION 22. ROLLE'S THEOREM…………………………………………….

ILLUSTRATION 23. MIDDLE VALUE THEOREM……………………………………

ILLUSTRATION 24. AVERAGE SPEED AND INSTANTANEOUS SPEED……….…

ILLUSTRATION 25. THE TOTAL INCOME VERSUS THE BEHAVIOR OF THE ELASTICITY OF THE DEMAND……………………………………………..………..

FOREWORD

The simplicity in the handling of mathematical concepts empowers the domain of knowledge with creative transfer in the complex development of the sciences

Alix Fuentes.

As an author, I am pleased to share some reflexive thoughts that led me to write math books for university students around the world, in order to overcome the paradigm that makes it difficult for many students to understand and use mathematics optimally. A product of this inspiration is this textbook hoping that the benefit of the author's academic and managerial experience will bring them great pleasure and satisfaction in the conceptual and applied interpretation of Differential Calculus.

In this context, this textbook is designed with academic alternatives focused on autonomous learning as an essential foundation of methodological development, where it is important to understand that the speed of learning depends only on the learner, because with this textbook you can learn at the speed that the student schedules, being careful to study each subject and make sure he or she has understood it well before continuing.

It is noteworthy that this textbook in its context allows the learner to learn to learn, learn by oneself and learn significantly. However, in the university environment other teaching aids and the academic and informatics resources offered by the valuable contributions of the professor allow the student to expand and deepen the knowledge acquired to the extent that it requires it.

In the study of Mathematics the learner should not skip subjects before mastering the previous contents, because Mathematic is easy to understand if one takes the time to comprehend the whole subject correctly before tackling a new related topic .

In general the study of mathematics requires having mastery in the previous fundamentals of each topic to understand a new topic. In this respect, this textbook offers in its development the basic contents required to understand and interpret each of the themes arranged in this textbook of Differential Calculus.

Consequently, if breaking paradigms is to create new opportunities in the development of knowledge and if this textbook allows the student to create new paradigms with pleasure and willingness towards Mathematics, then rewards will come in the mathematical logic applicable to all knowledge as one of the best academic achievements in Math that applies to all sciences.

INTRODUCTION

The textbook DIFFERENTIAL CALCULUS. A Mathematical Analysis for Applied Sciences is dedicated to all university students in distance or face-to-face programs that need to learn Differential Calculus for the purpose of using its contents in applied sciences.

Therefore, this work is oriented towards the applied sciences and offers the student an easy way to study mathematics without losing the conceptual depth nor the logical constructive development of the analytical processes proper to creative and reflexive thinking.

The relevant contributions of this textbook are centered on the Pedagogy for the Development of Autonomous Learning together with the logical and complete selection of topics that give the student the necessary knowledge to achieve the conceptual mastery of Differential Calculus in applied sciences, not including demonstrative processes or theoretical constructions typical of pure mathematics.

This structure includes the methodological design of autonomous learning tailored to the extent required by the learning of Differential Calculus to provide the student with a simple way to learn in the logical understanding of its contents. In this sense, it is essential to emphasize the conceptual and applied management of mathematical contents as added value in the construction and generation of knowledge applicable in the development of all sciences.

The student with mastery in the contents of Differential Calculus will be competent to apply them in any area of knowledge and if the learner requires to go more in depth in any of the treated topics, then the student will also be able to approach them with the sufficiency necessary to expand his or her knowledges.

The methodological design includes guidelines for autonomous learning to offer the student alternatives in learning to learn and learning by his or herself step by step in a logical-analytic-constructive process, proper to the disposition of the themes selected by the author. This logical order is required in each chapter, with the purpose of guiding the student to be an active part of the process in the construction of meaningful knowledge.

The contents of this textbook are structured in four main chapters focusing on the Differential Calculus program for applied sciences. Chapter One presents the theory of limits and continuity in functions as an essential fundament for understanding later the concept of the derivative of a function.

The conceptual management of the derivative of a function and the derivative of a function in a point offers in Chapter Two to the student the main arguments from which Differential Calculus is developed, offering the student another useful tool in the applied sciences.

With the conceptual domain of the derivative of a function we arrive at Chapter Three with the derivation rules for algebraic and transcendent functions. In this chapter the conceptual management of the derivative capitalizes on the results produced by the Pure Mathematics with the formulas that simplify demonstrative processes to be used with the benefits that expand the conceptualization of the derivative to all the algebraic and transcendent functions.

Chapter Four uses the results of previous chapters as working tools for problem solving in curves drawing, physics and mainly applications in the fields of economic and social sciences.

As part of the methodological design, Chapter Five is produced with the solutions to the recapitulation exercises proposed at the end of each of the previous chapters in order to provide feedback to the student in the fundamental contents of Differential Calculus.

Mathematics in general are demanding in the previous domain of mathematical academic fundamentals, which is the reason why the student must have studied and learned the previous knowledge arranged in the different academic prerequisites in this respect.

If for some reason these fundamentals are not part of the learner's domain, it is necessary that, in the measure of the progress achieved, the student will return to study the specific and required prerequisites in each subject.

In particular, if the student needs other mathematical bases in addition to those provided in this textbook, then the learner will also be able to find them and learn from the textbook BASIC MATH. An Introduction to Calculus; or ALGEBRA. A Mathematical Analysis Preliminary to Calculus.

These references are advisable considering that these previous books were designed by the author with the selection of contents necessary and sufficient to understand Calculus in the constructive and analytical process of the required thinking in the field of applied sciences.

The goal of the textbook DIFFERENTIAL CALCULUS. A Mathematical Analysis for Applied Sciences is to enable the student to learn to learn and learn for his or herself Differential Calculus as part of a logical-analytic-constructive process that involves the student in learning in a significant manner. This goal must be achieved from a positive attitude to the scope of learning proposed with enthusiasm and willingness to the benefits of Differential Calculus knowledge, overcoming the paradigms that might impede the diligence in the domain and meaningful understanding of Mathematics.

Chapter 1.  LIMITS AND CONTINUITY

CHAPTER ONE… CHAPTER ONE LIMITS AND CONTINUITY

SPECIFIC OBJECTIVES

bullet know the language of the Infinitesimal Calculus in the context of one of its branches called Differential Calculus.

bullet Recapitulate the fundamentals of the theory of limits required in the Differential Calculus to thereby provide the necessary basis for defining the derivative of a function.

bullet Use the necessary concepts of theory of limits in order to evaluate the trends of a function, continuity and discontinuity in mathematical functions.

Section 1.  GENERALITIES

GENERAL OBJECTIVE

Using the conceptual basis of the theory of limits in assessing trends of a function throughout its domain in order to analyze the continuity or discontinuity of the functions to develop a mathematical analysis of functional trends as a tool of working in Applied sciences.

INFINITESIMAL CALCULUS AND THE LIMITS THEORY

The Infinitesimal Calculus is the mathematics of the reason for change and the movement that reformulates some mathematical concepts through the use of limits.

Also the Infinitesimal Calculus expresses the physical laws and their consequences in mathematical terms, besides solving problems in the pure and applied sciences.

In this approach of Infinitesimal Calculus the concept of limit is relevant to construct new formulations in the development of Mathematics with Differential Calculus and Integral Calculus, among other mathematical contributions.

The infinitesimal Calculus has several branches. In this chapter are enunciated the following two branches:

Differential Calculus.

Integral Calculus.

In common terms, the Infinitesimal Calculus is called Calculus.

The Differential Calculus is a part of the Infinitesimal Calculus that is relevant to the mathematical analysis of the change of a function when the change in the variables is infinitesimal, that is, when this change tends to zero.

The main object of study of Differential Calculus is the derivative of a function with its uses and applications.

Differential Calculus has methods for finding how quickly a variable changes in value.

The graphical reference systems for the development of the Calculus are mainly the coordinated systems of Analytical Geometry, because in this textbook most of the applications of the Calculus are referred to movements and changes in the plane or space. In addition, it will be possible to observe some concepts of the Calculus that can be represented in geometric terms.

The theory of limits is a fundamental tool to develop the Differential Calculus from the definition of the derivative of a function.

The definition of the derivative makes use of the concept of limit of a function. For this reason, this chapter is developed with a conceptual synthesis of limits from the intuitive point of view without demonstrations with the purpose of using its conceptual management in the application of its mathematical formulations.

Therefore, in this textbook it is important to bear in mind that Differential Calculus uses the resulting from the mathematical rigor of definitions, properties and theorems of limit theory as part of the fundamental development of mathematical concepts Of the derivative.

These concepts are oriented to the study of the tendencies of a function, continuity, discontinuity and the derivative of a function.

For the purposes of this textbook, only the themes related to Differential Calculation are studied, focusing on the conceptual management of the derivative and its uses in the applied sciences.

Section 2.  LIMITS OF A FUNCTION

INTUITIVE LIMIT OF A FUNCTION

The intuitive limit of a function in Differential Calculus visually shows the trends of a function in a graphical system.

These tendencies are oriented to a function when the independent variable x approaches a specific number, or approaches a value that grows indefinitely towards plus infinite, or tends towards a value that decreases indefinitely to minus infinite.

In Advanced Calculus and Higher Mathematics programs the intuitive limit definition is formally approached with the mathematical rigor of the contributions of Augustin Louis Cauchy (1789 - 1857) and Karl Theodor Wilhelm Weierstrass (1815-1979), among others.

The theory of limits in this textbook includes the pedagogic process of significant systemic thinking from the significant reflection of the intuitive idea of a limit to the formal use of the properties of the limits that provide the tools to develop concepts of differential calculus in the Mathematics.

The conceptual basis of the theory of limits in this textbook is outside of the demonstration process, therefore in all approaches apply the concepts of the theory of limits with the respective verification and graphical interpretation without using the demonstrative rigor that is proper to the academic programs of Pure Mathematics.

Therefore the properties of limits are used in the calculation processes in the operationalization of the theory of limits.

Of course this chapter does not have the mathematical rigor in the demonstrations, but it does have the basic conceptual foundation necessary to understand the formal concept of the limit with which the concepts of continuity, discontinuity and the derivative of a function will be defined.

In order to understand the language in the limits theory, the symbology of the limit of a function is defined along with some examples that illustrate the tendencies of the functions in each case. With these illustrations are shown the intuitive definitions of the limits of a function.

SYMBOLOGY OF THE LIMIT OF A FUNCTION

Symbols are basic elements of communication. The symbology of the limit of a function has the basic structure that is shown in the following illustration.

ILLUSTRATION 1. SYMBOLOGY OF THE LIMIT OF A FUNCTION

SYMBOL

A symbol is represented by an ordered pair, where the first component is the signifier or object, and the second component is the meaning or idea; this is:

SYMBOL = (signifier, meaning) = (object, idea).

OBJECT: it is the recipient or signifier of the symbol by itself.

IDEA: is the meaning of the symbol.

MEANINGFUL REFLECTION

The symbology of the limit of a function is linked to the intuitive concept of the limit of a function in relation to the object and its meaning.

The limit of a function when the independent variable x tends towards a numerical value a could have several options, depending on the trend behavior of the function.

In this significant reflection, several considerations arise as follows:

(1) The limit of the function F(x) when x tends to the value of a numerical value a may result in a numerical value, or an infinitely large value, or an infinitely small value. These last values are indicated with the symbol of + ∞, or of -∞, respectively. Symbolically these considerations are indicated as follows:

Lim F(x) = L.

x→a

Lim F(x) = +∞.

x→a

Lim F(x) = -∞.

x→a

(2) The limit of a function F(x) when the independent variable x tends towards a numerical value a has the following considerations:

When x approaches the value of a to the left. This trend is indicated symbolically:

Lim F (x) = L.

X → a-

When x approaches the value of a to the right. This trend is indicated symbolically:

Lim F (x) = L.

X → a+

If the trend of the function F(x) is equal to the value of L when x tends to the value of a, both on the right and on the left, then the trend of the function F(x) will be indicated symbolically:

Lim F (x) = L.

x → a

(3) The limit of a function F(x) at a point x = a, considers that the function F(x) can be defined at the point x = a with value F(a); Or that the function F(x) is not defined in F(a).

(4) The limit of a function F(x) when the independent variable x tends towards an infinitely large or infinitely small value is indicated by the expression of + ∞ or -∞, depending on the case.

When the independent variable x tends towards infinitely large or infinitely small values at the limit of a function F(x), then one of the following alternatives may happen, as shown below:

Lim F(x) = L.

x→+∞

Lim F(x) = L.

x→-∞

Lim F(x) = +∞.

x→+∞

Lim F(x) = -∞.

x→+∞

Lim F(x) = -∞.

x→-∞

Lim F(x) = +∞.

x→-∞

Example 1.

Find graphically the trends of the function F(x) = 1/x, ∀ x ≠ 0. Indicate each of the trends of F(x) with the symbology of the limit of a function.

Solution:

The graph of the function F(x) = 1/x, ∀ x ≠ 0 is a curve having two branches in the Cartesian plane.

The first branch of the function F(x) = 1/x, ∀ x ≠ 0 defines the graph of the function F(x) = 1/x, ∀ x> 0.

The second branch of this curve is defined in the Cartesian plane with the graph of the function F (x) = 1/x, ∀ x < 0.

Therefore, with the information provided in the textbook Basic Mathematics to make the graph of this function is necessary to observe the following considerations:

x-axis is a horizontal asymptote of the function.

y-axis is a vertical function asymptote.

As the two axes of the Cartesian system are asymptotes of this function implies that the graph of the function F(x) = 1/x, ∀ x ≠ 0 does not intersect the x axis, and also intersects the y-axis.

The graph of the function F(x) shows four trends in the whole domain of the set of real numbers except for the value of x = 0.

The tendencies of the function are handled with the intuitive concept of limit of a function, and for these effects the symbology of the intuitive idea of limit is applied along with other necessary reflections necessary in this respect.

TABULATION

The tabulation of the function F(x) = 1/x, ∀ x ≠ 0 must have at least three points to plot each branch because in both cases corresponds to a curve.

For this example in order to better observe the trends of the function five points are considered for each branch as it is shown next in the following table:

Graphically the function F(x) = 1/x, ∀ x ≠ 0 shows the following trends of the function in the language of the limits of a function:

Lim F(x) = -∞ = Lim (1/x).

x → 0- x → 0-

It means: the value of the function F(x) = 1/x tends to minus infinity, when x tends to the value of zero from the left.

Lim F(x) = + ∞ = Lim (1/x).

x → 0+ x → 0+

It means: the value of the function F(x) = 1/x tends to plus infinity, when x tends to the value of zero by the right.

Lim F(x) = 0 = Lim (1/x).

x → + ∞ x → +∞

Means the value of the function F(x) = 1/x approaches zero as x approaches the value of plus infinity.

Lim F(x) = 0 = Lim (1/x).

x → -∞ x → -∞

It means: the value of the function F(x) = 1/x tends to zero, when x tends to the value of minus infinity.

It concludes:

The preceding behaviors of the function F(x) = 1/x, ∀ x ≠ 0 show an application of the intuitive concept of limit of a function.

It is observed in this respect that the concept of limit applied in this example illustrates the tendencies of a function in all its functional domain and in some cases in the points where it is not defined.

It is important to note that the function F(x) = 1/x when the variable x tends to the value of zero has two different limits. Because if the variable x approaches zero from the left the value of the function tends to minus infinity; but if the variable x approaches zero by the right the value of this function tends to plus infinity.

Then, it is conclusive to state that the function F(x) = 1/x, ∀ x ≠ 0 has no limit when the independent variable x approaches zero.

Other trends of the function are observed in the graph of the function F(x) = 1/x, ∀ x ≠ 0 when the variable x grows indefinitely towards plus infinity then the value of the corresponding limit of the function is zero; in contrast, when the variable x decreases indefinitely to minus infinity the value of this function tends to the value of zero.

GRAPHICAL BEHAVIOR

The graphical behavior of the function F(x) = 1/x; ∀ x ≠ 0 is as follows:

Example 2.

Graphically determine the trends of the function F(x) = 1/x² when the variable x tends to zero ∀ x ≠ 0. Express all other trends of this function in the symbolic language of the limit of a function.

Solution:

The graph of the function F(x) = 1/x², ∀ x ≠ 0 is a curve containing two branches in the Cartesian plane.

The first branch of the function F(x) = 1/x², ∀ x ≠ 0 defines the graph of the function F(x) = 1/x²; ∀ x > 0.

The second branch of this curve is defined in the Cartesian plane with the graph of the function F(x) = 1 / x²; ∀ x < 0.

The graph of the function F(x) = 1/x², ∀ x ≠ 0 must have for each of its two branches minimum three points in the corresponding tabulation, because the two branches of this function correspond to a curve.

Tabulation of the function F(x) = 1/x², ∀ x ≠ 0:

In order to facilitate the plotting of the graph and to better observe the trends of the function, we have calculated in this tabulation five points for each branch of the function F(x) = 1/x², ∀ x ≠ 0, as shown in the previous table of data.

GRAPHICAL BEHAVIOR



Graphically It is observed the trends of the function F(x) = 1/x², ∀ x ≠ 0 expressed in the symbolic language of the limit of a function as follows:

a. Limit of the function F(x) = 1/x ² , ∀ x ≠ 0 when the variable x tends to zero by the left and by the right.

Lim F(x) = +∞ = Lim (1/x²).

x→0- x→0-

It means: the value of the function F(x) = 1/x² tends to plus infinity, when x tends to the value of zero from the left.

Lim F(x) = +∞ = Lim (1/x²).

x→0+ x→0+

It means: the value of the function F(x) = 1/x² tends to plus infinity, when x tends to the value of zero by the right.

Lim F(x) = +∞ = Lim (1/x²) = Lim F(x) = +∞ = Lim (1/x²) = +∞.

x→0- x→0- x→0+ x→0+

It is concluded: that there exists the limit of the function F(x) = 1/x² when the variable x tends to zero with value of plus infinity; because this function tends to plus infinity when the variable x approaches zero by both the left and the right.

This limit is symbolized:

Lim F(x) = Lim (1/x²) = +∞.

x→0 x→0

b. The limit of the function F(x) = 1/x ² ∀ x ≠ 0 is zero, when the variable x tends to plus infinity. This tendency of the function is observed in the previous graph to conclude in the language of the limits theory the following:

Lim F(x) = 0 = Lim (1/x²).

x→+∞ x→+∞

c. Limit of the function F(x) = 1/x ² ∀ x ≠ 0 when the variable x tends to infinity. In theory of limits the tendency of this function is zero when the variable x tends toward minus infinity.

Symbolically it is:

Lim F(x) = 0 = Lim (1/x²).

x→-∞ x→-∞

Example 3.

Graphically displayed in the language of the theory of limits the trends of the quadratic function F(x) = x² when the independent variable x approaches from the right and left to the value of your domain at x = 0.

Solution:

The graph of the function F(x) = x² is a parabola that is open upwards with vertex at point V = (0, 0). This statement results in applying the knowledge studied in Basic Mathematics; therefore, the tabulation of the parabola F(x) = x² includes at least three points:

In order to illustrate the symmetry of this quadratic function with respect to the y-axis, the points chosen in the previous tabulation are symmetrical with respect to the y-axis.

GRAPHICAL BEHAVIOR

In the following graph it is observed that the function F(x) = x² tends to the value of zero when the variable x tends to zero by the right and by the left. Therefore, it is asserted that there exists the limit of the function F(x) = x² when the variable x tends to zero and the value of this limit is zero.



a. In the symbolic language of the theory of limits the trends of the quadratic function F(x) = x ² when the independent variable x approaches the value of your domain at x = 0 are:

Lim F(x) = Lim (x²) = 0.

x→0- x→0-

It means: the value of the function F(x) = x² tends to the value of zero, when x tends to the value of zero from the left.

Lim F(x) = Lim (x²) = 0.

x→0+ x→0+

It means: the value of the function F(x) = x² tends to zero, when x tends to the value of zero by the right.

It is concluded that the limit of the function F(x) = x² exists when the variable x tends to zero, because the tendency of this quadratic function is the same when the independent variable x tends to the zero value both by the left and right. This result is written as follows in the limit language of a function:

Lim x² = 0.

x→0

b. The limit of the function F(x) = x ² is plus infinite when the variable x tends to plus infinity. This trend of the function is observed in the previous graph with the following result:

Lim x² = +∞

x→+∞

c. The limit of the function F(x) = x ² is plus infinite when the variable x tends to minus infinity. In theory of limits this tendency is expressed symbolically:

Lim x² = +∞

x→-∞

With the analytical reflection of Example 1, Example 2 and Example 3, the intuitive applications of the limit concept are shown. This argument generates significant openness in understanding the following definitions of the limits of a function.

MEANINGFUL REFLECTION

In infinite limits there are four different forms in the possible results of the limit of a function F(x) when the independent variable x tends towards a certain value which belongs to the set of real numbers. These four different types are shown in the following table:

Other different forms in the possible results in calculating the limit of a function F(x) are shown in the following table:

DEFINITION OF LIMIT OF A FUNCTION

The formal limit definition of a function handles mathematical rigor both in the formulation and in the respective demonstrations proper to the development of theory of limits.

In this textbook the definitions of the limit of a function are oriented in the logical and analytical reflection of their conceptual components, which are the result of mathematical rigor. In this regard, the main focus is to achieve the learner get deep and simple conceptual understanding in the formulations, checks and graphical displays of particular behaviors of each case.

The theory of limits allows to understand among other mathematical concepts, the behavior and development in broad classes of functions of the superior mathematics; but in particular the purpose of this textbook is to provide the mathematical requirements to better understand the concept of the derivative of a function in Differential Calculus.

Thus, in the objectives of this section the formal definitions of the limits of a function are oriented to the mathematical arguments of the concept of limit to lead to a better understanding of the concept of the derivative of a function by the learner in the following chapter.

The formal definitions of limits of a function are arranged in three main approaches with variants in the significant formulation of their contents and these are referred to the following:

bullet CASE 1. LATERAL LIMITS AND THE LIMIT OF A FUNCTION IN A POINT. LIMIT OF A FUNCTION F(x) IS OF VALUE L, WHEN THE INDEPENDENT VARIABLE x TENDS TOWARD THE VALUE a.

bullet CASE 2. INFINITE LIMITS. LIMIT OF ONE FUNCTION F(x) IS INFINITE WHEN THE INDEPENDENT VARIABLE x TENDS TOWARDS THE ∞.

bullet CASE 3. LIMITS IN THE INFINITE. LIMIT OF A FUNCTION F(x) IS OF VALUE L, WHEN THE INDEPENDENT VARIABLE x TENDS TOWARDS THE ∞.

Next, case 1 defines that if F(x) has a limit of value L when the independent variable x tends toward a, this limit L will be unique.

CASE 1.  LATERAL LIMITS AND LIMITS OF A FUNCTION ON A POINT



DEFINITION OF LATERAL LIMITS AND LIMIT OF A FUNCTION IN A POINT… DEFINITION OF LATERAL LIMITS AND LIMIT OF A FUNCTION IN A POINT If F(x) tends towards a number L when the independent variable x tends towards the value of a, both by the right and on the left, then L will be called the limit of the function F(x) at the point of x = a. This definition is symbolized: Lim F(x) = Lim F(x) = L ➜ Lim F(x) = L. x→a- x→a+ x→a The mathematical rigor of this definition of lateral limits and the existence of the limit of a function F(x) when the variable x tends towards a point a includes the handling of the following observations: F(x) is defined for all the values of the independent variable x which are subject to considerations: ∃ c ∈ ℝ ∧ b ∈ ℝ ⎮ x ∈ (c, a) ∧ x ∈ (a, b) ∧ c < a < b. The value of a is not always defined in the domain of the function F(x). If ε is a positive number, then it is necessary to find a positive number δ so that the following is true: In the expression 0 < ⎮x - a⎮ < δ must be fulfilled that ⎮F(x) - L⎮ < ε. The above result means that for every positive value of ε there is a number δ > 0 such that for all values of x the above inequalities are satisfied.

The concept of limit expressed in case 1 offers the learner mathematical arguments to determine when the limit of a function exists.

This information is vital in the constructive process of differential calculus to understand the concept of the derivative of a function. In order to visualize this conceptual management the following illustration is presented:

Lim F(x) = Lim F(x) = L ➜ Lim F(x) = L.

x→a- x→a+ x→a

ILLUSTRATION 2. GRAPHICAL INTERPRETATION OF THE LATERAL LIMITS AND THE LIMIT OF A FUNCTION IN A POINT



Example 4.

Show the geometric meaning of the equality of the next limit, and apply the definition of the limit