Mathematical analysis in clear language. Fundamentals of higher mathematics. The concept of limit in mathematics

Mathematical analysis for dummies. Lesson 1. Sets.

Concept of set

A bunch of is a collection of certain objects. What sets can there be? Firstly, finite or infinite. For example, a set of matches in a box is a finite set; they can be taken and counted. It is much more difficult to count the number of grains of sand on the beach, but in principle it is possible. And this quantity is expressed by some finite number. So there are a lot of grains of sand on the beach too, of course. But the set of points on a straight line is infinite. Since, firstly, the straight line itself is infinite and you can put as many points on it as you like. The set of points on a line segment is also infinite. Because theoretically the point can be as small as desired. Of course, we physically cannot draw a point that is, for example, smaller than the size of an atom, but from a mathematical point of view, a point has no size. Its size is zero. What happens when you divide a number by zero? That's right, infinity. And although the set of points on a straight line and on a segment tends to infinity, they are not the same thing. A set is not a quantity of something, but a collection of some objects. And only those sets that contain absolutely identical objects are considered equal. If one set contains the same objects as another set, but plus one more “left” object, then these are no longer equal sets.

Let's look at an example. Let us have two sets. The first is the collection of all points on a line. The second is the set of all points on a line segment. Why are they not equal? Firstly, the segment and the straight line may not even intersect. Then they are certainly not equal, since they contain completely different points. If they intersect, then they have only one common point. Everyone else is just as different. What if the segment lies on a straight line? Then all points of the segment are also points of the line. But not all points on a line are points on a segment. So in this case, the sets cannot be considered equal (the same).

Each set is defined by a rule that uniquely determines whether an element belongs to this set or not. What might these rules be? For example, if the set is finite, you can stupidly list all its objects. You can set a range. For example, all integers from 1 to 10. This will also be a finite set, but here we are not listing its elements, but formulating a rule. Or inequality, for example, all numbers greater than 10. This will be an infinite set, since it is impossible to name the most big number- no matter what number we name, there is always this number plus 1.

As a rule, sets are designated by capital letters of the Latin alphabet A, B, C, and so on. If a set consists of specific elements and we want to define it as a list of these elements, then we can enclose this list in curly braces, for example A=(a, b, c, d). If a is an element of the set A, then it is written as follows: a Î A. If a is not an element of the set A, then write a Ï A. One of the important sets is the set N of all natural numbers N=(1,2,3,...,) . There is also a special, so-called empty set, which does not contain a single element. The empty set is denoted by the symbol Æ .

Definition 1 (definition of equality of sets). Sets A and B are equal if they consist of the same elements, that is, if xÎ A follows x Î B and vice versa, from x Î B follows x Î A.

Formally, the equality of two sets is written as follows:

(A=B) := " x (( x Î A ) Û (x Î B )),

This means that for any object x the relations xÎ A and xО B are equivalent.

Here " – universal quantifier (" xreads "for everyone" x").

Definition 2 (subset definition). A bunch of A is a subset of the set IN, if any X belonging to the multitude A, belongs to the set IN. Formally, this can be represented as an expression:

(A Ì B) := " x((x Î A) Þ (x Î B))

If A Ì B but A ¹ B, then A is a proper subset of the set IN. As an example, we can again cite a straight line and a segment. If a segment lies on a line, then the set of its points is a subset of the points of this line. Or, another example. The set of integers that are evenly divisible by 3 is a subset of the set of integers.

Comment. The empty set is a subset of any set.

Set Operations

The following operations are possible on sets:

An association. The essence of this operation is to combine two sets into one containing elements of each of the combined sets. Formally it looks like this:

C=AÈ B: = {x:x Î A or xÎ B}

Example. Let's solve the inequality | 2 x+ 3 | > 7.

It follows either the inequality 2x+3 >7, for 2x+3≥0, then x>2

or inequality 2x+3<-7, для 2x+3 <0, тогда x<-5.

The set of solutions to this inequality is the union of the sets (-∞,-5) È (2, ∞).

Let's check. Let's calculate the value of the expression | 2 x+ 3 | for several points lying and not lying in a given range:

As you can see, everything was resolved correctly (the boundary ranges are indicated in red).

Intersection. Intersection is the operation of creating a new set from two, containing elements that are included in both of these sets. To visualize this, let's imagine that we have two sets of points on the plane, namely figure A and figure B. Their intersection denotes figure C - this is the result of the operation of intersection of sets:

Formally, the operation of intersection of sets is written as follows:

C=A Ç B:= (x: x Î A and x О B )

Example. Let us have a set Then C=A Ç B = {5,6,7}

Subtraction. Subtraction of sets is the exclusion from the subtrahend of those elements that are contained in the subtrahend and the subtractor:

Formally, subtraction of a set is written as follows:

A\B:={x:x Î A and xÏ B}

Example. May we have plenty A=(1,2,3,4,5,6,7), B=(5,6,7,8,9,10). Then C=A\ B = { 1,2,3,4}

Addition. Complement is a unary operation (an operation not on two, but on one set). This operation is the result of subtracting a given set from the complete universal set (the set that includes all other sets).

A : = (x:x Î U and x Ï A) = U \ A

Graphically this can be represented as:

Symmetrical difference. In contrast to the usual difference, with a symmetric difference of sets, only those elements that are present in either one or the other set remain. Or, in simple terms, it is created from two sets, but those elements that are in both sets are excluded from it:

Mathematically this can be expressed as follows:

A D B:= (A\B) È ( B\A) = (A È B) \ (A Ç B)

Properties of operations on sets.

From the definitions of union and intersection of sets it follows that the operations of intersection and union have the following properties:

1. Commutativity.

A È B=BÈ A
AÇ B=BÇ A

1. Associativity.

(A È B) È C=AÈ ( B È C)
(A Ç B) Ç C=AÇ ( B Ç C)

Limits give all mathematics students a lot of trouble. To solve a limit, sometimes you have to use a lot of tricks and choose from a variety of solution methods exactly the one that is suitable for a particular example.

In this article we will not help you understand the limits of your capabilities or comprehend the limits of control, but we will try to answer the question: how to understand limits in higher mathematics? Understanding comes with experience, so at the same time we will give several detailed examples of solving limits with explanations.

The concept of limit in mathematics

The first question is: what is this limit and the limit of what? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since this is what students most often encounter. But first, the most general definition of a limit:

Let's say there is some variable value. If this value in the process of change unlimitedly approaches a certain number a , That a – the limit of this value.

For a function defined in a certain interval f(x)=y such a number is called a limit A , which the function tends to when X , tending to a certain point A . Dot A belongs to the interval on which the function is defined.

It sounds cumbersome, but it is written very simply:

Lim- from English limit- limit.

There is also a geometric explanation for determining the limit, but here we will not delve into the theory, since we are more interested in the practical rather than the theoretical side of the issue. When we say that X tends to some value, this means that the variable does not take on the value of a number, but approaches it infinitely close.

Let's give a specific example. The task is to find the limit.

To solve this example, we substitute the value x=3 into a function. We get:

By the way, if you are interested, read a separate article on this topic.

In examples X can tend to any value. It can be any number or infinity. Here's an example when X tends to infinity:

Intuitively, the larger the number in the denominator, the smaller the value the function will take. So, with unlimited growth X meaning 1/x will decrease and approach zero.

As you can see, to solve the limit, you just need to substitute the value to strive for into the function X . However, this is the simplest case. Often finding the limit is not so obvious. Within the limits there are uncertainties of the type 0/0 or infinity/infinity . What to do in such cases? Resort to tricks!

Uncertainties within

Uncertainty of the form infinity/infinity

Let there be a limit:

If we try to substitute infinity into the function, we will get infinity in both the numerator and the denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: you need to notice how you can transform the function in such a way that the uncertainty goes away. In our case, we divide the numerator and denominator by X in the senior degree. What will happen?

From the example already discussed above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:

To resolve type uncertainties infinity/infinity divide the numerator and denominator by X to the highest degree.

By the way! For our readers there is now a 10% discount on

Another type of uncertainty: 0/0

As always, substituting values ​​into the function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that we have a quadratic equation in the numerator. Let's find the roots and write:

Let's reduce and get:

So, if you are faced with type uncertainty 0/0 – factor the numerator and denominator.

To make it easier for you to solve examples, we present a table with the limits of some functions:

L'Hopital's rule within

Another powerful way to eliminate both types of uncertainty. What is the essence of the method?

If there is uncertainty in the limit, take the derivative of the numerator and denominator until the uncertainty disappears.

L'Hopital's rule looks like this:

Important point : the limit in which the derivatives of the numerator and denominator stand instead of the numerator and denominator must exist.

And now - a real example:

There is typical uncertainty 0/0 . Let's take the derivatives of the numerator and denominator:

Voila, uncertainty is resolved quickly and elegantly.

We hope that you will be able to usefully apply this information in practice and find the answer to the question “how to solve limits in higher mathematics.” If you need to calculate the limit of a sequence or the limit of a function at a point, and there is absolutely no time for this work, contact a professional student service for a quick and detailed solution.

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:

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

What to do with uncertainty of the form: $ \bigg [\frac(0)(0) \bigg ] $

Let's push the limit in the last two examples to infinity and consider the uncertainty: $ \bigg [\frac(\infty)(\infty) \bigg ] $

Algorithm for calculating limits

So, let's briefly summarize the examples and create an algorithm for solving the limits:

1. 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.
2. 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.
3. 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!

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!

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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...