A heap of terrible formulas, manuals on higher mathematics that you open and immediately close, the painful search for a solution to a seemingly very simple problem .... This situation is not uncommon, especially when a math textbook was last opened in the distant 11th grade. Meanwhile, in universities, the curricula of many specialties provide for the study of everyone's favorite higher mathematics. And in this situation, you often feel like a complete teapot in front of a heap of terrible mathematical gibberish. Moreover, a similar situation can arise in the study of any subject, especially from the cycle of natural sciences.

What to do? For a full-time student, everything is much simpler, unless, of course, the subject is not very neglected. You can consult a teacher, classmates, and just write off from a neighbor on the desk. Even a full teapot in higher mathematics will survive the session in such scenarios.

And if a person is studying at the correspondence department of a university, and higher mathematics, to put it mildly, is unlikely to be required in the future? In addition, there is no time for classes. So it is, in most cases, so, but no one canceled the performance of tests and passing the exam (most often, written). With tests in higher mathematics, everything is easier, whether you are a teapot or not a teapot - math test can be ordered. For example, I have. Other items can be ordered as well. Not here anymore. But the implementation and submission of test papers for review will not yet lead to the coveted entry in the grade book. It often happens that a work of art, made to order, needs to be defended, and it is necessary to explain why that formula follows from these letters. In addition, exams are coming, and there you will already have to solve determinants, limits and derivatives INDEPENDENTLY. Unless, of course, the teacher does not accept valuable gifts, or there is no hired well-wisher outside the classroom.

Let me give you some very important advice. At tests, exams in exact and natural sciences, IT IS VERY IMPORTANT TO UNDERSTAND SOMETHING. Remember, AT LEAST SOMETHING. The complete absence of thought processes just infuriates the teacher, I know of cases when part-time students were wrapped up 5-6 times. I remember that one young man passed the test 4 times, and after each retake he turned to me for a free warranty consultation. In the end, I noticed that in the answer he wrote the letter “pe” instead of the letter “pi”, which was followed by severe sanctions from the reviewer. The student DID NOT EVEN WANT TO LOOK into the assignment, which he casually rewrote

You can be a complete dummy in higher mathematics, but it is highly desirable to know that the derivative of a constant is equal to zero. Because if you answer some stupidity to an elementary question, then there is a high probability that your studies at the university will end for you. Teachers are much more favorable to the student who AT LEAST TRYING to understand the subject, to the one who, albeit mistakenly, but tries to solve, explain or prove something. And this statement is true for all disciplines. Therefore, the position “I know nothing, I understand nothing” should be resolutely rejected.

The second important advice is to ATTEND LECTURES, even if there are not many of them. I already mentioned this on the main page of the site. Mathematics for correspondence students. It makes no sense to repeat why it is VERY important, read there.

So, what to do if there is a test on the nose, an exam in higher mathematics, and things are deplorable - the state of a full, or rather, empty teapot?

One option is to hire a tutor. The largest database of tutors can be found (mainly Moscow) or (mainly St. Petersburg). Using a search engine, it is quite likely to find a tutor in your city, or look at local advertising newspapers. The price for the services of a tutor can vary from 400 or more rubles per hour, depending on the qualifications of the teacher. It should be noted that cheap does not mean bad, especially if you have a good mathematical background. At the same time, for 2-3K rubles you will get a LOT. In vain no one takes such money, and in vain no one pays such money ;-). The only important point - try to choose a tutor with a specialized pedagogical education. And in fact, we do not go to the dentist for legal help.

Recently, online tutoring service is gaining popularity. It is very convenient when you need to urgently 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 megacities.

In the course of higher mathematics, it is very difficult to master some things without a tutor, you just need a “live” explanation.

Nevertheless, it is quite possible to understand 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 work experience in my specialty.

Let's start to rake mathematical gibberish. It's okay, even if you are a teapot, higher mathematics is really simple and really accessible.

And you need to start by repeating the school course of mathematics. Repetition is the mother of pain.

Before you begin to study my methodological materials, and in general begin to study any materials in higher mathematics, I HIGHLY RECOMMEND that you read the following.

In order to successfully solve problems in higher mathematics, you MUST:

GET A MICROCALCULATOR.

Of the programs - Excel (an excellent choice!). I uploaded the manual for "dummies" to the library.

There is? Already good.

– From the rearrangement of the terms - the sum does not change: .
But these are completely different things:

It is simply impossible to rearrange "x" and "four". At the same time, we recall the iconic letter "x", which in mathematics means an unknown or variable value.

– By rearranging the factors - the product does not change: .
With division, such a trick will not work, and these are two completely different fractions, and rearranging the numerator with the denominator does not do without consequences.
We also recall that the multiplication sign (“dots”) is most often not written:,

– Recall the rules for expanding brackets:
- here the signs of the terms do not change
- and here they are reversed.
And for multiplication:

In general, it suffices to remember that TWO MINUS GIVES A PLUS, a THREE MINUS - GIVE MINUS. And, try not to get confused in this when solving problems in higher mathematics (a very frequent and annoying mistake).

– Recall the reduction of like terms, You should have a good understanding of the following operation:

– Remember what a degree is:

, , , .

A degree is just an ordinary multiplication.

–Remember that fractions can be reduced: (reduced by 2), (reduced by five), (reduced by ).

– Remember actions with fractions:

and also, a very important rule for reducing fractions to a common denominator:

If these examples are not clear, see school textbooks.
Without this, it will be TOUGH.

ADVICE: all INTERMEDIATE calculations in higher mathematics are best done in ORDINARY RIGHT AND IRREGULAR FRACTIONS, even if scary fractions like . This fraction SHOULD NOT be represented as , and, moreover, DO 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’s just better to write or.

– The equation. It has a left side and a right side. For example:

You can transfer 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:

New Page 1

Mathematical analysis for dummies. Lesson 1. Sets.

The concept of a set

A bunch of is a collection of some objects. What can be sets? First, finite or infinite. For example, the set of matches in a box is a finite set, they can be taken and counted. The number of grains of sand on the beach is much more difficult to count, but, in principle, possible. And this quantity is expressed by some finite number. So many grains of sand on the beach, of course. But the set of points on a straight line is an infinite set. Since, firstly, the 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 a point can be arbitrarily small. Of course, we cannot physically draw a point, for example, smaller than the size of an atom, but, from the point of view of mathematics, 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, this is not the same thing. A set is not a quantity of something there, but a collection of any objects. And only those sets that contain exactly the same 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.

Consider an example. Let's say we have two sets. The first is the collection of all points on the line. The second is the set of all points on a straight line segment. Why are they not equal? First, a line segment and a 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. All the rest are 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 line segment. So in this case, the sets cannot be considered equal (identical).

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 enumerate 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 do not list its elements, but formulate a rule. Or inequality, for example, all numbers are greater than 10. This will already be an infinite set, since it is impossible to name the largest number - no matter what number we call, there is always this number plus 1.

As a rule, sets are denoted by capital letters of the Latin alphabet A, B, C, and so on. If the 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 this 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 BUT and B are equal if they consist of the same elements, that is, if from xн A follows x н B and vice versa, from x н B follows x н 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 " is the universal quantifier (" xreads "for each x").

Definition 2 (subset definition). A bunch of BUT is a subset of the set AT if any X belonging to the set BUT, belongs to the set AT. Formally, this can be expressed 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 AT. As an example, again, a straight line and a segment can be cited. 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.

Operations on sets

The following operations are possible on sets:

Union. 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 implies 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 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 the given range:

x	| 2 x+ 3 |
-10	17
-6	9
-5	7
-4	5
-2	1
0	3
1	5
2	7
3	9
5	13

As you can see, everything was decided correctly (the border ranges are marked in red).

intersection. Intersection is the operation of creating a new set of 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. Set subtraction is the exclusion from the subtracted set 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 many 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 the given set from the complete universal set (the set that includes all other sets).

A := (x:x О U and x П 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 either in one or in another 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)

A heap of terrible formulas, manuals on higher mathematics that you open and immediately close, the painful search for a solution to a seemingly very simple problem .... This situation is not uncommon, especially when a math textbook was last opened in the distant 11th grade. Meanwhile, in universities, the curricula of many specialties provide for the study of everyone's favorite higher mathematics. And in this situation, you often feel like a complete teapot in front of a heap of terrible mathematical gibberish. Moreover, a similar situation can arise in the study of any subject, especially from the cycle of natural sciences.

What to do? For a full-time student, everything is much simpler, unless, of course, the subject is not very neglected. You can consult a teacher, classmates, and just write off from a neighbor on the desk. Even a full teapot in higher mathematics will survive the session in such scenarios.

And if a person is studying at the correspondence department of a university, and higher mathematics, to put it mildly, is unlikely to be required in the future? In addition, there is no time for classes. So it is, in most cases, so, but no one canceled the performance of tests and passing the exam (most often, written). With tests in higher mathematics, everything is easier, whether you are a teapot or not a teapot - math test can be ordered. For example, I have. Other items can be ordered as well. Not here anymore. But the implementation and submission of test papers for review will not yet lead to the coveted entry in the grade book. It often happens that a work of art, made to order, needs to be defended, and it is necessary to explain why that formula follows from these letters. In addition, exams are coming, and there you will already have to solve determinants, limits and derivatives INDEPENDENTLY. Unless, of course, the teacher does not accept valuable gifts, or there is no hired well-wisher outside the classroom.

Let me give you some very important advice. At tests, exams in exact and natural sciences, IT IS VERY IMPORTANT TO UNDERSTAND SOMETHING. Remember, AT LEAST SOMETHING. The complete absence of thought processes just infuriates the teacher, I know of cases when part-time students were wrapped up 5-6 times. I remember that one young man passed the test 4 times, and after each retake he turned to me for a free warranty consultation. In the end, I noticed that in the answer he wrote the letter “pe” instead of the letter “pi”, which was followed by severe sanctions from the reviewer. The student DID NOT EVEN WANT TO LOOK into the assignment, which he casually rewrote

You can be a complete dummy in higher mathematics, but it is highly desirable to know that the derivative of a constant is equal to zero. Because if you answer some stupidity to an elementary question, then there is a high probability that your studies at the university will end for you. Teachers are much more favorable to the student who AT LEAST TRYING to understand the subject, to the one who, albeit mistakenly, but tries to solve, explain or prove something. And this statement is true for all disciplines. Therefore, the position “I know nothing, I understand nothing” should be resolutely rejected.

The second important advice is to ATTEND LECTURES, even if there are not many of them. I already mentioned this on the main page of the site. Mathematics for correspondence students. It makes no sense to repeat why it is VERY important, read there.

So, what to do if there is a test on the nose, an exam in higher mathematics, and things are deplorable - the state of a full, or rather, empty teapot?

One option is to hire a tutor. The largest database of tutors can be found (mainly Moscow) or (mainly St. Petersburg). Using a search engine, it is quite likely to find a tutor in your city, or look at local advertising newspapers. The price for the services of a tutor can vary from 400 or more rubles per hour, depending on the qualifications of the teacher. It should be noted that cheap does not mean bad, especially if you have a good mathematical background. At the same time, for 2-3K rubles you will get a LOT. In vain no one takes such money, and in vain no one pays such money ;-). The only important point - try to choose a tutor with a specialized pedagogical education. And in fact, we do not go to the dentist for legal help.

Recently, online tutoring service is gaining popularity. It is very convenient when you need to urgently 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 megacities.

In the course of higher mathematics, it is very difficult to master some things without a tutor, you just need a “live” explanation.

Nevertheless, it is quite possible to understand 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 work experience in my specialty.

Let's start to rake mathematical gibberish. It's okay, even if you are a teapot, higher mathematics is really simple and really accessible.

And you need to start by repeating the school course of mathematics. Repetition is the mother of pain.

Before you begin to study my methodological materials, and in general begin to study any materials in higher mathematics, I HIGHLY RECOMMEND that you read the following.

In order to successfully solve problems in higher mathematics, you MUST:

GET A MICROCALCULATOR.

Of the programs - Excel (an excellent choice!). I uploaded the manual for "dummies" to the library.

There is? Already good.

– From the rearrangement of the terms - the sum does not change: .
But these are completely different things:

It is simply impossible to rearrange "x" and "four". At the same time, we recall the iconic letter "x", which in mathematics means an unknown or variable value.

– By rearranging the factors - the product does not change: .
With division, such a trick will not work, and these are two completely different fractions, and rearranging the numerator with the denominator does not do without consequences.
We also recall that the multiplication sign (“dots”) is most often not written:,

– Recall the rules for expanding brackets:
- here the signs of the terms do not change
- and here they are reversed.
And for multiplication:

In general, it suffices to remember that TWO MINUS GIVES A PLUS, a THREE MINUS - GIVE MINUS. And, try not to get confused in this when solving problems in higher mathematics (a very frequent and annoying mistake).

– Recall the reduction of like terms, You should have a good understanding of the following operation:

– Remember what a degree is:

, , , .

A degree is just an ordinary multiplication.

–Remember that fractions can be reduced: (reduced by 2), (reduced by five), (reduced by ).

– Remember actions with fractions:

and also, a very important rule for reducing fractions to a common denominator:

If these examples are not clear, see school textbooks.
Without this, it will be TOUGH.

ADVICE: all INTERMEDIATE calculations in higher mathematics are best done in ORDINARY RIGHT AND IRREGULAR FRACTIONS, even if scary fractions like . This fraction SHOULD NOT be represented as , and, moreover, DO 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’s just better to write or.

– The equation. It has a left side and a right side. For example:

You can transfer 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:

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

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

The concept of a limit in mathematics

The first question is: what is the 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 it is with them that students most often encounter. But first, the most general definition of a limit:

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

For a function defined in some interval f(x)=y the limit is the number A , to which the function tends 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 the definition of the limit, but here we will not go into theory, since we are more interested in the practical 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 take a concrete example. The challenge 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 in basic operations on matrices, read a separate article on this topic.

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

It is intuitively clear that the larger the number in the denominator, the smaller the value will be taken by the function. So, with unlimited growth X meaning 1/x will decrease and approach zero.

As you can see, in order 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 type 0/0 or infinity/infinity . What to do in such cases? Use 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 both in the numerator and in 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 is gone. In our case, we divide the numerator and denominator by X in senior degree. What will happen?

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

To uncover type ambiguities 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 any kind of work

Another type of uncertainty: 0/0

As always, substitution into the value 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 encounter type ambiguity 0/0 - factorize the numerator and denominator.

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

L'Hopital's rule within

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

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

Visually, L'Hopital's rule looks like this:

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

And now a real example:

There is a typical uncertainty 0/0 . Take the derivatives of the numerator and denominator:

Voila, the uncertainty is eliminated quickly and elegantly.

We hope that you will be able to put this information to good use 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 no time for this work from the word “absolutely”, contact a professional student service for a quick and detailed solution.

The Calculus category contains free online video lessons on this topic. Mathematical analysis is a set of branches of mathematics that study 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 (TFKP), 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 the calculus of variations. Learning calculus from video lessons will be useful for both beginners and more experienced mathematicians. You can watch video lessons from the section Mathematical analysis for free at any convenient time. Some video lessons on mathematical analysis have additional materials that can be downloaded. Happy learning!

Total materials: 12
Shown Materials: 1-10

What is the derivative of a function

Do you want to know what is the derivative of a function in mathematics? Of course, you 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 I will explain the meaning of the derivative on the fingers so that even a round teapot can understand. But first, you better watch my previous video, where I also talk about the function in an accessible way. In this video tutorial, we are simple, clear and illustrative 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 the power of sets. This question concerns the quantitative characterization of sets. If the set is finite, then we can talk about the number of its elements. But what about infinite sets? Indeed, in this case there will be no concept of more or less. To solve this problem, such a concept as power is introduced. Power is a tool for quantitatively comparing infinite sets. This lesson gives...

Limit of a function at a point - definition, examples

This online lesson talks about such a concept as the limit of a function at a point - definition, examples. Most elements of the study of functions are based on the basic concept of the limit of a function. Here, the limit of a function at a point will be considered 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 the graph for better assimilation of the material. This lesson also looks at other examples and provides a rigorous definition of one-sided...

Convergence of power series - an example of how to find the area of ​​​​convergence, research

This video tutorial talks about such a concept as the 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 area of ​​convergence is all values ​​of the variable x for which the corresponding numerical series converge. For research, you can use the d'Alembert test and use it to show that the power series converges or diverges, and when ...

What is primitive

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 we just have to dot the i's. The antiderivative is the "parent" function for the derivative. Finding an 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 for a given mother. We are now making the transition from...

The geometric meaning of the derivative

In this video I will talk about the geometric meaning of the derivative. You will learn that the geometric meaning of the derivative is that the derivative and the slope of the tangent are almost the same thing. I say "almost" because the derivative is equal to the tangent of the slope of the tangent. We can assume that the derivative and the slope of the tangent are closely related. If the slope is large, then the derivative is also large, and the function at this point increases sharply. If the angle of inclination is small, then the derivative is also small...

What is a function in mathematics

Want to know what a function is in mathematics? In this video tutorial, we will simply and clearly, using graphic illustrations and illustrative life examples, tell you what a function is, what its argument is, what functions are (increasing, decreasing, mixed), how you can set 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 relationship 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 are limits at infinity. Most of the elementary functions are defined for an arbitrarily large value of the argument. In this case, it is important to know the behavior of the function at infinity. One element of the study of such behavior is to find the limit of the function at infinity. Although infinity is not a number, and no point on the number line corresponds to it, the definition of the limit on ...