Limits in mathematics for dummies: explanation, theory, examples of solutions.
The Mathematical Analysis category contains free online video lessons on this topic. Mathematical analysis is a set of branches of mathematics that deal with the study of functions and their generalizations using the methods of differential and integral calculus. These include: functional analysis, including the theory of the Lebesgue integral, complex analysis (CFCA), which studies functions defined on the complex plane, the theory of series and multidimensional integrals, non-standard analysis, which studies infinitesimal and infinitely large numbers, vector analysis, and calculus of variations. Learning mathematical analysis from video lessons will be useful for both beginners and more experienced mathematicians. You can watch video lessons from the Mathematical Analysis section for free at any time. Some video lessons on mathematical analysis come with additional materials that can be downloaded. Enjoy your learning!
Total materials: 12
Materials shown: 1-10
What is the derivative of a function
Want to know what the derivative of a function is in mathematics? You, of course, have heard about the derivative many times and even, probably, took this very derivative at school, completely not understanding the meaning of your actions. In this video, I will not teach you formulas, but will explain the meaning of the derivative on your fingers in such a way that even a round teapot can understand it. But first, you better watch my previous video, where I also talk about the function in an accessible way. In this video tutorial we will use simple, clear and clear life examples...
Introduction to analysis. Power of sets
Online lesson “Introduction to analysis. Power of Sets” is devoted to the question of such a concept as cardinality of sets. This question concerns the quantitative characteristics of sets. If the set is finite, then we can talk about the number of its elements. But what about infinite sets? After all, in this case there will be no concept of more or less. To solve this problem, the concept of power is introduced. Power is a tool for quantitative comparison of infinite sets. This lesson provides...
Limit of a function at a point - definition, examples
This online lesson talks about the concept of the limit of a function at a point - definition, examples. Most elements of function research rely on the basic concept of the limit of a function. Here we will consider the limit of a function at a point using a simple example, after which a strict definition of the limit of a function at a point will be given with a detailed illustration on a graph for better understanding of the material. This lesson also covers other examples and sets out a strict definition of one-sided...
Convergence of power series - an example of how to find the region of convergence, research
This video lesson talks about the concept of convergence of power series, an example of how to find the area of convergence, research. A power series is a special case of a functional series when its members are power functions of the argument x. The region of convergence represents all values of the variable x for which the corresponding number series converge. For research, you can use d’Alembert’s test and use it to show that a power series converges or diverges, and when...
What is an antiderivative
In this video I will tell you about the antiderivative, which is a close relative of the derivative. In fact, you already know almost everything about her if you watched my previous videos, and all we have to do is dot the i’s. The antiderivative is the “parent” function for the derivative. Finding the antiderivative means answering the question: whose child is it? If the daughter is known, then we must find the mother. Previously, on the contrary, we were looking for a daughter based on a given mother. Now we are making the reverse transition - from...
Geometric meaning of derivative
In this video I will talk about the geometric meaning of derivatives. You will learn that the geometric meaning of the derivative is that the derivative and the angle of inclination of the tangent are almost the same thing. I say “almost” because the derivative is equal to the tangent of the tangent angle. We can assume that the derivative and the slope of the tangent are closely related. If the angle of inclination is large, then the derivative is large, and the function at this point increases sharply. If the angle of inclination is small, then the derivative is small...
What is a function in mathematics
Want to know what a function is in mathematics? In this video lesson, we will explain simply and clearly, using graphic illustrations and clear life examples, what a function is, what its argument is, what functions there are (increasing, decreasing, mixed), how you can define a function (using a graph, table, formulas). You will see that a relationship that shows how one quantity is related to another quantity is called a function. Any function is a connection between quantities...
Limit of a function at infinity - definition, examples
The lesson “Limit of a function at infinity - definition, examples” is devoted to the question of what limits at infinity are. Most elementary functions are defined for arbitrarily large argument values. In this case, it is important to know the behavior of the function at infinity. One element of studying this behavior is to find the limit of the function at infinity. Although infinity is not a number and there is no point on the number line corresponding to it, the definition of a limit on...
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Theory.
NEW. Natanzon S.M. Short course in mathematical analysis. 2004 98 pp. djvu. 1.2 MB.
This publication is a brief recording of a course of lectures given by the author for 1st year students of the Independent Moscow University in the 1997-1998 and 2002-2003 academic years.
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NEW. E.B. Boronina. Mathematical analysis. Lecture notes. 2007 160 pp. pdf. 2.1 MB.
This book is written for students of technical universities who want to prepare for the exam in mathematical analysis. The content of this book fully corresponds to the program for the course “Mathematical Analysis”, an exam for which is provided in most higher educational institutions of Russia. The program helps you quickly and without unnecessary difficulties find the necessary answer to the question posed.
The questions were compiled by the author based on personal experience, taking into account the requirements of teachers.
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Arkhipov, Sadovnichy, Chubarikov. Lectures on mathematical analysis. Textbook.analysis. 1999 635 pp. djvu. 5.2 MB.
The book is a textbook for a course in mathematical analysis and is devoted to differential and integral calculus of functions of one and several variables. It is based on lectures given by the authors at the Faculty of Mechanics and Mathematics of Moscow State University. M. V. Lomonosov. The textbook proposes a new approach to the presentation of a number of basic concepts and theorems of analysis, as well as to the course content itself. For students of universities, pedagogical universities and universities with in-depth study of mathematics
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Aksenov A.P. Mathematical analysis. (Fourier series. Fourier integral. Summation of divergent series.) Textbook. 1999 86 pages PDF 1.2 Mb.
The manual complies with the state standard of the discipline "Mathematical Analysis" in the direction of bachelor's training 510200 "Applied Mathematics and Computer Science".
Contains a presentation of theoretical material in accordance with the current program on the topics: "Fourier Series", "Fourier Integral", "Summation of Divergent Series". A large number of examples are given. The application of the Cesaro and Abel-Poisson methods in the theory of series is outlined. The question of harmonic analysis of functions given empirically is considered.
Intended for students of the Faculty of Physics and Mechanics of specialties 010200, 010300, 071100, 210300, as well as for teachers conducting practical classes.
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Aksenov. Mathematical analysis. (Integrals depending on a parameter. Double integrals. Curvilinear integrals.) Textbook St. Petersburg. year 2000. 145 pp. PDF. Size 2.3 MB. djvu.
The manual complies with the state standard of the discipline "Mathematical Analysis" in the direction of bachelor's training 510200 "Applied Mathematics and Computer Science". Contains a presentation of theoretical material in accordance with the current program on the topics: “Integrals depending on a parameter, proper and improper”, “Double integral”, “Curvilinear integrals of the first and second kind”, “Calculation of the areas of curved surfaces specified both explicitly and parametric equations", "Eulerian integrals (Beta function and Gamma function)". A large number of examples and problems have been analyzed (47 in total).
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De Bruyne. Asymptotic methods in analysis. 245 pp. djvu. 1.6 MB.
The book contains an elementary presentation of a number of methods used in analysis to obtain asymptotic formulas. The importance of the methods presented in the book, the clarity and accessibility of the presentation make this book very valuable for anyone starting to get acquainted with such methods. The book is of undoubted interest also for those who are already familiar with this area of analysis.
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Stefan Banach. Differential and integral calculus. 1966 437 pp. djvu. 7.7 MB.
Stefan Banach is one of the greatest mathematicians of the 20th century. This book was conceived by him as a guide for initial familiarization with the subject. Meanwhile, the author managed in a small book to masterfully cover almost all the basic material of differential and integral calculus, without scaring off the reader with the scrupulous rigor of the presentation.
The book is distinguished by its simplicity and brevity of presentation. It contains many well-chosen examples, as well as problems for independent solution. Designed for students of colleges (especially correspondence), teacher training institutes, as well as engineering and technical workers who wish to refresh their memory of the basic facts of differential and integral calculus.
When preparing the second edition, the experience of teaching this book in some higher technical educational institutions was taken into account; In this regard, a small number of additions have been made to the book, and some places in the text have been corrected. This brought the book closer to the level of modern textbooks on mathematical analysis and made it possible to use it in colleges and universities.
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B.M. Budak, S.V. Fomin. Multiple integrals and series. Textbook. 1965. 606 pp. djvu. 4.6 MB.
For physics and mathematics university faculties.
I RECOMMEND!!!. Especially for PHYSICISTS.
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Viosagmir I.A. Higher mathematics for dummies. Function limit. 2011. 95 pp. pdf. 6.1 MB.
I welcome you to my first book on the limits of function. This is the first part of my upcoming series “higher mathematics for dummies”. The title of the book should already tell you a lot about it, but you may completely misunderstand it. This book is dedicated not to “dummies,” but to all those who find it difficult to understand what professors do in their books. I'm sure you understand me. I myself was and am in such a situation that I am simply forced to read the same sentence several times. This is fine? I think no.
So what makes my book different from all the others? Firstly, the language here is normal, not “abstruse”; secondly, there are a lot of examples discussed here, which, by the way, will probably be useful to you; thirdly, the text has a significant difference from each other - the main things are highlighted with certain markers, and finally, my goal is only one - your understanding. Only one thing is required from you: desire and skills. “Skills?” - you ask. Yes! Ability to remember and understand.
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V.N. Gorbuzov. Mathematical analysis: integrals depending on parameters. Uch. allowance. 2006 496 pp. PDF. 1.6 MB.
The differential and integral calculus of functions defined by certain improper integrals, which depend on parameters, is presented. Designed for university students studying mathematics and physics, as well as for students of technical specialties with an extended program in mathematics.
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Dorogovtsev A.Ya. Mathematical analysis. A short course in modern presentation. Second edition. 2004 560 pp. djvu. 5.1 MB.
The book contains a brief and at the same time quite complete presentation of the modern course in mathematical analysis. The book is intended primarily for students of universities and technical universities and is intended for initial study of the course. A modernized presentation of a number of sections is given: functions of several variables, multiple integrals, integrals over manifolds, the Stokes formula and others are explained. The theoretical material is illustrated by a large number of exercises and examples. . For university students, mathematics teachers, engineering and technical workers.
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Egorov V.I., Salimova A.F. Definite and multiple integrals. Elements of field theory. 2004 256 pp. djvu. 1.6 MB.
The publication presents the theory and basic applications of definite and multiple integrals, as well as elements of field theory. The material is adapted to the modern program of mathematical education in higher technical educational institutions, and for use in computer teaching systems. The book is intended for students of technical universities. It may also be useful for teachers, engineers, and scientists.
Clearly a well-written book. All statements of the theory are illustrated with examples. I recommend it as additional literature for understanding the material.
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Evgrafov. Asymptotic estimates and entire functions. 320 pp. djvu. 3.2 MB.
The book is devoted to the presentation of various methods of asymptotic estimates (Laplace's method, saddle point method, residue theory) used in the theory of entire functions. The methods are illustrated mainly based on the material of this theory. The basic facts from the theory of entire functions are not assumed to be known to the reader - their presentation is organically included in the structure of the book. A chapter on the asymptotics of conformal mappings has been added to the 3rd edition. The book is intended for a wide range of readers - from students to scientists, both mathematicians and applied scientists.
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I WOULD. Zeldovich, I.M. Yaglom. Higher mathematics for beginning physicists and technicians. 1982 514 pp. djvu. 12.3 MB.
This book is an introduction to mathematical analysis. Along with the presentation of the principles of analytical geometry and mathematical analysis (differential and integral calculus), the book contains concepts about power and trigonometric series and the simplest differential equations, and also touches on a number of sections and topics from physics (mechanics and theory of oscillations, theory of electrical circuits, radioactive decay , lasers, etc.). The book is intended for readers interested in natural science applications of higher mathematics, university and college teachers, as well as future physicists and engineers.
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Zeldovich, Yaglom. The book is in three parts: 1. Elements of higher mathematics. Contains: Functions and graphs (50 pages), What is a derivative (50 pages), What is an integral (20 pages), Calculation of derivatives (20 pages), Integration techniques (20 pages), Series, simplest differential equations (35 pages), Study of functions, several problems in geometry (55 pages) 2. Applications of higher mathematics to certain questions of physics and technology (160 pages) Contains: Radioactive decay and nuclear fission, Mechanics, Vibrations, Thermal motion of molecules, distribution of air density in the atmosphere, Absorption and emission of light, lasers, Electric circuits and oscillatory motions in them 3. Additional topics from higher mathematics (50 pages) Contains: Complex numbers, What functions does a physicist need, The wonderful Dirac delta function, Some applications of the function of a complex variable and delta functions. 4. Applications, Answers, Directions, Solutions. Did you catch it, what kind of book? You can go nuts just by reading the table of contents. But this is not a textbook on mathematics, THIS BOOK IS ABOUT HOW TO USE MATH. By the way, by studying it, you will inevitably learn physics too. Super. djvu, 500 pages. Size 8.7 MB.
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Zorich V.A. Mathematical analysis. In 2 parts. Textbook. 1 - 1997, 2 - 1984. 567+640 pp. djvu. 9.6+7.4 MB.
University textbook for students of physics and mathematics. It may be useful for students of faculties and universities with advanced mathematical training, as well as specialists in the field of mathematics and its applications. The book reflects the connection of the course of classical analysis with modern mathematical courses (algebra, differential geometry, differential equations, complex and functional analysis).
The first part included: an introduction to analysis (logical symbolism, set, function, real number, limit, continuity); differential and integral calculus of a function of one variable; differential calculus of functions of several variables.
The second part of the textbook includes the following sections: Multidimensional integral. Differential forms and their integration. Series and integrals depending on a parameter (including series and Fourier transforms, as well as asymptotic expansions).
Problem solving aids.
NEW. Sadovnichaya I.V., Khoroshilova E.V. Definite integral: theory and practice of calculations. 2008 528 pp. djvu. 2.7 MB.
The publication is devoted to the theoretical and practical aspects of calculating definite integrals, as well as methods of their evaluation, properties and applications to solving various geometric and physical problems. The book contains sections devoted to methods for calculating proper integrals, properties of improper integrals, geometric and physical applications of a certain integral, as well as some generalizations of the Riemann integral - the Lebesgue and Stieltjes integrals.
The presentation of theoretical material is supported by a large number (more than 220) of analyzed examples of calculations, estimates and studies of the properties of certain integrals; at the end of each paragraph there are problems for independent solution (more than 640, the vast majority with solutions).
The purpose of the manual is to help the student while going through the topic “Definite Integral” in lectures and practical classes. The student can contact him to obtain background information on the issue that has arisen. The book can also be useful to teachers and anyone wishing to study this topic in sufficient detail and broadly.
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NEW. Khoroshilova E.V. Mathematical analysis: indefinite integral. (to help with practical exercises). 2007 184 pp. djvu. 822 KB.
The book provides basic theoretical information about indefinite integrals, considers most of the well-known techniques and methods of integration and various classes of integrable functions (indicating methods of integration). The presentation of the material is supported by a large number of analyzed examples of calculating integrals (more than 200 integrals), at the end of each paragraph there are problems for independent solution (more than 200 problems with answers).
The manual contains the following paragraphs: “The concept of an indefinite integral”, “Basic methods of integration”, “Integration of rational fractions”, “Integration of irrational functions”, “Integration of trigonometric functions”, “Integration of hyperbolic, exponential, logarithmic and other transcendental functions”. The book is intended for mastering the theory of indefinite integral in practice, developing skills in practical integration, consolidating the course of lectures, using it at seminars and while preparing homework. The purpose of the manual is to help the student master various techniques and methods of integration.
For university students, including those majoring in mathematics, studying integral calculus as part of a course in mathematical analysis.
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NEW. V.F. Butuzov, N.Ch. Krutitskaya, G.N. Medvedev, A.A. Shishkin. Mathematical analysis in questions and problems: Proc. allowance. 5th ed., rev. 2002 480 pp. djvu. 3.8 MB.
The manual covers all sections of the course on mathematical analysis of functions of one and several variables. For each topic, basic theoretical information is briefly outlined and test questions are proposed; solutions to standard and non-standard problems are provided; Tasks and exercises are given for independent work with answers and instructions. Fourth edition 2001
For university students.
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A.A. Burtsev. Methods for solving exam problems in mathematical analysis, 2nd semester, 1st year. 2010 pdf, 56 pp. 275 Kb.
Variants of problems for four previous ones. of the year.
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Vinogradova I. A. et al. Problems and exercises in mathematical analysis (part 1). 1988 djvu, 416 pp. 5.0 MB.
The collection is compiled on the material of classes in the course of mathematical analysis in the first year of the Faculty of Mechanics and Mathematics of Moscow State University and reflects the teaching experience of the department of mathematical analysis. It consists of two parts, corresponding to the I and II semester. Each part contains separate computational exercises and theoretical problems. The first part includes sketching graphs of functions, calculating limits, differential calculus of functions of one real variable, and theoretical problems. The second part is the indefinite integral, the definite Riemann integral, differential calculus of functions of many variables, theoretical problems. In chapters containing computational exercises, each paragraph is preceded by detailed methodological instructions. They give all the definitions used in this section, the formulation of the main theorems, the derivation of some necessary relations, provide detailed solutions to typical problems, and draw attention to common errors. Most of the problems and exercises are different from the problems contained in the well-known problem book by B. P. Demidovich. Both parts of the collection include about 1800 calculation exercises and 350 theoretical problems.
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Vinogradova I. A. et al. Problems and exercises in mathematical analysis (part 2). 1991 djvu, 352 pp. 3.2 MB.
The problem book corresponds to the course of mathematical analysis taught in the second year and contains the following sections: double and triple integrals and their geometric and physical applications, curvilinear and surface integrals of the first and second kind. The necessary theoretical information, typical algorithms suitable for solving entire classes of problems are provided, and detailed methodological instructions are given.
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Vinogradov and others. Ed. Sadovnichigo. Problems and exercises in mathematical analysis. 51 pp. PDF. 1.9 MB.
The section on plotting graphs is discussed in great detail. The considered examples occupy 35 pages.
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Zheltukhin. Indefinite integrals: calculation methods. 2005 year. Size 427 KB. PDF, 80 pages. Useful guide, can be used as a reference. It not only introduces all the methods for calculating integrals, but also provides a lot of examples for each rule. I recommend.
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Zaporzhets. Guide to solving problems in mathematical analysis. 4th ed. 460 pp. djvu. 7.7 MB.
Covers all sections from studying functions to solving differential equations. Useful book.
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Kalinin, Petrova, Kharin. Indefinite and definite integrals. 2005 year. 230 pp. PDF. 1.2 MB.
Finally, mathematicians began to write books for physicists and other technical students, and not for themselves. I recommend it if you want to learn how to calculate, rather than prove, lemmas and theorems.
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Kalinin, Petrova. Multiple, curvilinear and surface integrals. Tutorial. 2005 year. 230 pp. PDF. 1.2 MB.
This manual provides examples of calculating various integrals.
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Kaplan. Practical classes in higher mathematics. Analytical geometry, differential calculus, integral calculus, integration of differential equations. In 2 files in one archive. General 925 pp. djvu. 6.9 MB.
Examples of problem solving throughout the general mathematics course are considered.
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K.N. Lungu, etc. Collection of problems in higher mathematics. Part 2 for 2nd year. 2007 djvu, 593 pp. 4.1 Mb.
Series and integrals. Vector and complex analysis. Differential equations. Probability theory. Operational calculus. This is not just a problem book, but also a tutorial. You can use it to learn how to solve problems.
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Lungu, Makarov. Higher mathematics. Guide to problem solving. Part 1. 2005. Size 2.2 MB. djvu, 315 pp.
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I.A. Maroon. Differential and integral calculus in examples and problems (Functions of one variable). 1970 djvu. 400 pp. 11.3 MB.
The book is a guide to solving problems of mathematical analysis (functions of one variable). Contains brief theoretical introductions, solutions to typical examples and problems for independent solution. In addition to problems of an algorithmic-computational nature, it contains many tasks that illustrate the theory and contribute to its deeper assimilation, developing students’ independent mathematical thinking. The purpose of the book is to teach students to independently solve problems in the course of mathematical analysis
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D.T. Writing. Higher mathematics 100 exam questions. 1999 djvu. 304 pp. 9.3 MB.
This manual is intended primarily for students preparing to take the exam in higher mathematics in the 1st year. It contains answers to the oral exam questions presented in a concise, accessible form. The manual can be useful for all categories of students studying higher mathematics to one degree or another. It contains the necessary material for 10 sections of the higher mathematics course, which are usually studied by students in the first year of a university (technical school). Answers to 108 exam questions (with subparagraphs - much more) are usually accompanied by solutions to relevant examples and problems.
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Sobol B.V., Mishnyakov N.T., Porksheyan V.M. Workshop on higher mathematics. 2006 630 pp. djvu. 5.4 MB.
The book includes all sections of the standard course of higher mathematics for a wide range of specialties of higher educational institutions.
Each chapter (the corresponding section of the course) contains reference material, as well as the basic theoretical principles necessary to solve problems. A distinctive feature of this publication is a large number of problems with solutions, which allows it to be used not only for classroom teaching, but also for students’ independent work. The problems are presented by topic and systematized by solution methods. Each chapter ends with sets of tasks for independent solution, equipped with answers.
The completeness of the presentation of the material and the relative compactness of this publication make it possible to recommend it to teachers and students of higher educational institutions, as well as students of advanced training institutes who wish to systematize their knowledge and skills on this subject.
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E.P. Sulyandziga, G.A. Ushakova. MATHEMATICS TESTS: LIMIT, DERIVATIVE, ELEMENTS OF ALGEBRA AND GEOMETRY. Uch. allowance. year 2009. pdf, 127 pp. 1.1 Mb.
The proposed tutorial can be considered as a collection of tasks. The problems cover traditional topics - the basics of mathematical analysis: a function, its limit and derivative. There are problems on the basics of linear algebra and analytical geometry. Since the limit and derivative of a function are more difficult, and in addition, these topics are fundamental to integral calculus, the greatest attention is paid to them: solutions to typical problems are analyzed in detail. The material collected in the textbook was repeatedly used in practical classes.
For first year students of all universities.
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For those who want to learn how to find limits, in this article we will tell you about it. We won’t delve into the theory; teachers usually give it at lectures. So the “boring theory” should be jotted down in your notebooks. If this is not the case, then you can read textbooks taken from the library of the educational institution or from other Internet resources.
So, the concept of limit is quite important in the study of higher mathematics, especially when you come across integral calculus and understand the connection between limit and integral. The current material will look at simple examples, as well as ways to solve them.
Examples of solutions
| Example 1 |
| Calculate a) $ \lim_(x \to 0) \frac(1)(x) $; b)$ \lim_(x \to \infty) \frac(1)(x) $ |
| Solution |
|
a) $$ \lim \limits_(x \to 0) \frac(1)(x) = \infty $$ b)$$ \lim_(x \to \infty) \frac(1)(x) = 0 $$ People often send us these limits with a request to help solve them. We decided to highlight them as a separate example and explain that these limits just need to be remembered, as a rule. If you cannot solve your problem, then send it to us. We will provide detailed solution. You will be able to view the progress of the calculation and gain information. This will help you get your grade from your teacher in a timely manner! |
| Answer |
| $$ \text(a)) \lim \limits_(x \to 0) \frac(1)(x) = \infty \text( b))\lim \limits_(x \to \infty) \frac(1 )(x) = 0 $$ |
What to do with uncertainty of the form: $ \bigg [\frac(0)(0) \bigg ] $
| Example 3 |
| Solve $ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) $ |
| Solution |
|
As always, we start by substituting the value $ x $ into the expression under the limit sign. $$ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) = \frac((-1)^2-1)(-1+1)=\frac( 0)(0) $$ What's next now? What should happen in the end? Since this is uncertainty, this is not an answer yet and we continue the calculation. Since we have a polynomial in the numerators, we will factorize it using the formula familiar to everyone from school $$ a^2-b^2=(a-b)(a+b) $$. Do you remember? Great! Now go ahead and use it with the song :) We find that the numerator $ x^2-1=(x-1)(x+1) $ We continue to solve taking into account the above transformation: $$ \lim \limits_(x \to -1)\frac(x^2-1)(x+1) = \lim \limits_(x \to -1)\frac((x-1)(x+ 1))(x+1) = $$ $$ = \lim \limits_(x \to -1)(x-1)=-1-1=-2 $$ |
| Answer |
| $$ \lim \limits_(x \to -1) \frac(x^2-1)(x+1) = -2 $$ |
Let's push the limit in the last two examples to infinity and consider the uncertainty: $ \bigg [\frac(\infty)(\infty) \bigg ] $
| Example 5 |
| Calculate $ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) $ |
| Solution |
|
$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) = \frac(\infty)(\infty) $ What to do? What should I do? Don't panic, because the impossible is possible. It is necessary to take out the x in both the numerator and the denominator, and then reduce it. After this, try to calculate the limit. Let's try... $$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) =\lim \limits_(x \to \infty) \frac(x^2(1-\frac (1)(x^2)))(x(1+\frac(1)(x))) = $$ $$ = \lim \limits_(x \to \infty) \frac(x(1-\frac(1)(x^2)))((1+\frac(1)(x))) = $$ Using the definition from Example 2 and substituting infinity for x, we get: $$ = \frac(\infty(1-\frac(1)(\infty)))((1+\frac(1)(\infty))) = \frac(\infty \cdot 1)(1+ 0) = \frac(\infty)(1) = \infty $$ |
| Answer |
| $$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) = \infty $$ |
Algorithm for calculating limits
So, let's briefly summarize the examples and create an algorithm for solving the limits:
- Substitute point x into the expression following the limit sign. If a certain number or infinity is obtained, then the limit is completely solved. Otherwise, we have uncertainty: “zero divided by zero” or “infinity divided by infinity” and move on to the next steps of the instructions.
- To eliminate the uncertainty of “zero divided by zero,” you need to factor the numerator and denominator. Reduce similar ones. Substitute point x into the expression under the limit sign.
- If the uncertainty is “infinity divided by infinity,” then we take out both the numerator and the denominator x to the greatest degree. We shorten the X's. We substitute the values of x from under the limit into the remaining expression.
In this article, you learned the basics of solving limits, often used in the Calculus course. Of course, these are not all types of problems offered by examiners, but only the simplest limits. We'll talk about other types of assignments in future articles, but first you need to learn this lesson in order to move forward. Let's discuss what to do if there are roots, degrees, study infinitesimal equivalent functions, remarkable limits, L'Hopital's rule.
If you can't figure out the limits yourself, don't panic. We are always happy to help!
A pile of scary formulas, manuals on higher mathematics that you open and immediately close, a painful search for a solution to a seemingly simple problem... This situation is not uncommon, especially when the last time a mathematics textbook was opened was in the distant 11th grade. Meanwhile, in universities, the curricula of many specialties include the study of everyone’s favorite higher mathematics. And in this situation, you often feel like a complete teapot in front of a pile of terrible mathematical gobbledygook. Moreover, a similar situation can arise when studying any subject, especially from the natural sciences.
What to do? For a full-time student, everything is much simpler, unless, of course, the subject is very neglected. You can consult with the teacher, classmates, or simply copy from your neighbor at your desk. Even a full teapot in higher mathematics will survive the session in such situations.
What if a person is studying in the correspondence department of a university, and higher mathematics, to put it mildly, is unlikely to be required in the future? Besides, there is absolutely no time for classes. This is how it is, in most cases, but no one canceled the completion of tests and passing the exam (most often, written). With tests in higher mathematics, everything is simpler, whether you are a dummy or not - Mathematics test can be ordered. For example, for me. You can also order for other items. No longer here. But completing and submitting tests for review will not lead to the coveted entry in the grade book. It often happens that a custom-made work of art needs to be protected and explained why those letters lead to that formula. In addition, exams are coming up, and there you will have to solve determinants, limits and derivatives BY YOURSELF. Unless, of course, the teacher accepts valuable gifts, or there is a hired well-wisher outside the classroom.
Let me give you very important advice. During tests and exams in exact and natural sciences, IT IS VERY IMPORTANT TO UNDERSTAND AT LEAST SOMETHING. Remember, AT LEAST SOMETHING. The complete lack of thought processes simply infuriates the teacher; I know of cases where part-time students were turned away 5-6 times. I remember one young man took the test 4 times, and after each retake he turned to me for a free guarantee consultation. In the end, I noticed that in his answer he wrote the letter “pe” instead of the letter “pi”, for which severe sanctions followed from the reviewer. The student DID NOT EVEN WANT TO GET INTO the assignment, which he carelessly rewrote
You can be a complete novice in higher mathematics, but it is extremely desirable to know that the derivative of a constant is equal to zero. Because if you answer some stupid question to a basic question, then there is a high probability that your studies at the university will end there. Teachers are much more favorable towards the student who AT LEAST TRYING to understand the subject, to the one who, albeit mistakenly, is trying to solve, explain or prove something. And this statement is true for all disciplines. Therefore, the attitude “I don’t know anything, I don’t understand anything” should be resolutely rejected.
The second important tip is to ATTEND LECTURES, even if they are few. I already mentioned this on the main page of the site. Mathematics for correspondence students. There is no point in repeating why this is VERY important, read there.
So, what to do if a test or an exam in higher mathematics is just around the corner, but things are deplorable - a state of a full, or, more precisely, empty teapot?
One option is to hire a tutor. The largest database of tutors can be found in (mainly Moscow) or (mainly St. Petersburg). Using a search engine, it is quite possible to find a tutor in your city, or look at local advertising newspapers. The price of a tutor's services can vary from 400 rubles or more per hour, depending on the qualifications of the teacher. It should be noted that cheap does not mean bad, especially if you have good mathematical training. At the same time, for 2-3K rubles you will get a LOT. It’s in vain that no one takes that kind of money, and it’s in vain that no one pays that kind of money ;-). The only important point is to try to choose a tutor with specialized pedagogical education. And in fact, we don’t go to the dentist for legal help.
Recently, online tutoring services have been gaining popularity. It is very convenient when you urgently need to solve one or two problems, understand a topic, or prepare for an exam. The undoubted advantage is the prices, which are several times lower than those of an offline tutor + saving time on travel, which is especially important for residents of big cities.
In a higher mathematics course, it is very difficult to master some things without a tutor; you need a “live” explanation.
However, it is quite possible to figure out many types of problems on your own, and the purpose of this section of the site is to teach you how to solve typical examples and problems that are almost always found in exams. Moreover, for a number of tasks there are “hard” algorithms, where there is “no escape” from the correct solution. And, to the best of my knowledge, I will try to help you, especially since I have a pedagogical education and experience in my specialty.
Let's start clearing out the mathematical gobbledygook. It’s okay, even if you are a beginner, higher mathematics is really simple and really accessible.
And you need to start by repeating the school mathematics course. Repetition is the mother of torment.
Before you begin to study my teaching materials, and indeed begin to study any materials on higher mathematics, I STRONGLY RECOMMEND that you read the following.
In order to successfully solve problems in higher mathematics, you MUST:
STOCK UP WITH A MICRO CALCULATOR.
Programs include Excel (great choice!). I uploaded the manual for dummies to the library.
Eat? Already good.
– Rearranging the terms does not change the sum.: .
But these are completely different things:
You can’t just rearrange “X” and “four”. At the same time, let’s remember the iconic letter “X,” which in mathematics denotes an unknown or variable quantity.
– Rearranging the factors does not change the product: .
This trick will not work with division, and these are two completely different fractions and rearranging the numerator with the denominator does not do without consequences.
We also remember that it is most often customary not to write the multiplication sign (“dot”): ,
– Remember the rules for opening parentheses:
– here the signs of the terms do not change
- and here they change to the opposite.
And for multiplication: 
In general, it is enough to remember that TWO MINUSES GIVE A PLUS, A THREE MINUSES – GIVE A MINUS. And try NOT to get confused about this when solving problems in higher mathematics (a very common and annoying mistake).
– Let us recall the reduction of similar terms, You should understand the following action well:
– Let's remember what a degree is:
, , 
, 
.
A power is just a simple multiplication.
–Remember that fractions can be reduced: (reduced by 2), (reduced by five), (reduced by ).
– Recalling operations with fractions:
and also, a very important rule for bringing fractions to a common denominator: 
If these examples are unclear, look at school textbooks.
Without this it will be TIGHT.
ADVICE: it is better to carry out all INTERMEDIATE calculations in higher mathematics in ORDINARY PROPER AND IMPROPER FRACTIONS, even if you get terrible fractions like . This fraction should NOT be represented in the form , and, moreover, you should NOT divide the numerator by the denominator on the calculator, getting 4.334552102….
The EXCEPTION to the rule is the final answer of the task, then it is better to write down or.
– The equation. It has a left side and a right side. For example:
You can move any term to another part by changing its sign:
Let's move, for example, all the terms to the left side:
Or to the right:
Matrix called a rectangular table filled with numbers. The most important characteristics of a matrix are the number of rows and the number of columns. If a matrix has the same number of rows and columns, it is called square. Matrices are designated in capital Latin letters.
The numbers themselves are called matrix elements and characterize their position in the matrix by specifying the row number and column number and writing them in the form of a double index, with the row number being written first and then the column number. For example, a 14 is a matrix element located in the first row and fourth column, a 32 is in the third row and second column.
The main diagonal of a square matrix call elements that have the same indices, that is, those elements whose row number coincides with the column number. Side diagonal runs “perpendicular” to the main diagonal.
Of particular importance are the so-called unit matrices. These are square matrices with 1 on the main diagonal and all other numbers equal to 0. Unit matrices are denoted by E. The matrices are called equal, if they have the same number of rows, number of columns, and all elements having the same indices are equal. The matrix is called null, if all its elements are equal to 0. The zero matrix is denoted O.
The simplest operations with matrices
1. Multiplying a matrix by a number. To do this, you need to multiply each element of the matrix by a given number.
2. Addition of matrices. You can only add matrices of the same size, that is, having the same number of rows and the same number of columns. When adding matrices, their corresponding elements are added together.
3. Matrix transposition. When a matrix is transposed, its rows become columns and vice versa. The resulting matrix is called transposed and is denoted by A T. The following properties hold for transposing matrices:
4. Matrix multiplication. The following properties exist for a product of matrices:
- You can multiply matrices if the number of columns of the first matrix is equal to the number of rows of the second matrix.
- The result is a matrix whose number of rows is equal to the number of rows of the first matrix, and the number of columns is equal to the number of columns of the second matrix.
- Matrix multiplication is non-commutative. This means that if the matrices in the product are rearranged, the result changes. Moreover, if you can calculate the product A∙B, this does not mean at all that you can calculate the product B∙A.
- Let C = A∙B. To determine the matrix element C located in i-that line and k-that column you need to take i-that row of the first matrix to be multiplied and k-th column is second. Next, take the elements of these rows and columns one by one and multiply them. We take the first element from the row of the first matrix and multiply it by the first element of the column of the second matrix. Next, we take the second row element of the first matrix and multiply it by the second column element of the second matrix, and so on. And then all these works must be added up.
Matrix determinant
Determinant (determinant) square matrix A is a number denoted by det A, less often | A| or simply Δ, and is calculated in a certain way. For a 1x1 matrix, the determinant is the single element of the matrix itself. For a 2x2 matrix, the determinant is found using the following formula:
Minors and algebraic complements
Consider matrix A. Let us choose in it s lines and s columns. Let's create a square matrix of elements located at the intersection of the resulting rows and columns. Minor matrix A of order s is called the determinant of the resulting matrix.
Consider a square matrix A. In it we choose s lines and s columns. Additional minor to minor order s is called a determinant made up of the elements remaining after crossing out the given rows and columns.
Algebraic complement to element a ik of a square matrix A is the additional minor to this element multiplied by (–1) i+k, Where i+k is the sum of the row and column numbers of the element a ik. Denotes the algebraic complement of A ik.
Calculation of the determinant of a matrix through algebraic additions
Consider a square matrix A. To calculate its determinant, you need to select any of its rows or columns and find the product of each element of this row or column by its algebraic complement. And then we need to sum up all these works.
The calculation of an algebraic complement can be reduced to the calculation of a determinant of size more than 2x2. In this case, such a calculation also needs to be carried out through algebraic additions, and so on until the algebraic additions that need to be calculated become 2x2 in size, then use the formula above.
inverse matrix
Consider a square matrix A. The matrix A –1 is called reverse to matrix A if their products are equal to the identity matrix. The inverse matrix exists only for square matrices. An inverse matrix exists only if matrix A non-degenerate, that is, its determinant is not equal to zero. Otherwise, it is impossible to calculate the inverse matrix. To construct the inverse matrix you need:
- Find the determinant of the matrix.
- Find the algebraic complement for each element of the matrix.
- Construct a matrix from algebraic additions and be sure to transpose it. Transposition is often forgotten.
- Divide the resulting matrix by the determinant of the original matrix.
Thus, if matrix A has a size of 3x3, its inverse matrix has the form:
Derivative
Let's consider some function f(x), depending on the argument x. Let this function be defined at the point x 0 and some of its surroundings, is continuous at this point and its surroundings. Let's consider a small change in the argument of the function ∆ x. Let the function change to ∆ f(x). Then derivative of a function at this point the following relation is called.
