For those who want to learn how to find the limits in this article we will talk about it. We will not delve into the theory, it is usually given in lectures by teachers. So the "boring theory" should be outlined in your notebooks. If this is not the case, then you can read textbooks taken from the library of the educational institution or on other Internet resources.

So, the concept of the limit is quite important in the study of the course of higher mathematics, especially when you come across the integral calculus and understand the relationship between the limit and the integral. In the current material, simple examples will be considered, as well as ways to solve them.

Solution examples

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)$$ \lim_(x \to \infty) \frac(1)(x) = 0 $$ We often get these limits sent to us asking for help to solve. We decided to highlight them as a separate example and explain that these limits simply need to be remembered, as a rule. If you cannot solve your problem, then send it to us. We will provide a detailed solution. You will be able to familiarize yourself with the progress of the calculation and gather information. This will help you get a credit from the teacher in a timely manner!
Answer
$$ \text(a)) \lim \limits_(x \to \to 0) \frac(1)(x) = \infty \text( b))\lim \limits_(x \to \infty) \frac(1 )(x) = 0 $$

What to do with the 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 of $ 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? What should be the result? Since this is an uncertainty, this is not yet an answer and we continue the calculation. Since we have a polynomial in the numerators, we decompose it into factors using the familiar formula $$ a^2-b^2=(a-b)(a+b) $$. Remembered? Fine! Now go ahead and apply it with the song :) We get that the numerator $ x^2-1=(x-1)(x+1) $ We continue to solve given 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 take 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? How to be? Do not panic, because the impossible is possible. It is necessary to take out the brackets in both the numerator and the denominator X, and then reduce it. After that, try to calculate the limit. Trying... $$ \lim \limits_(x \to \infty) \frac(x^2-1)(x+1) =\lim \limits_(x \to \infty) \frac(x^2(1-\frac (1)(x^2)))(x(1+\frac(1)(x))) = $$ $$ = \lim \limits_(x \to \infty) \frac(x(1-\frac(1)(x^2)))((1+\frac(1)(x))) = $$ 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 analyzed examples and make an algorithm for solving the limits:

Substitute point x in the expression following the limit sign. If a certain number is obtained, or infinity, then the limit is completely solved. Otherwise, we have uncertainty: "zero divided by zero" or "infinity divided by infinity" and proceed to the next paragraphs of the instruction.
To eliminate the uncertainty "zero divide by zero" you need to factorize the numerator and denominator. Reduce similar. Substitute the point x in the expression under the limit sign.
If the uncertainty is "infinity divided by infinity", then we take out both in the numerator and in the denominator x of the greatest degree. We shorten the x's. We substitute x values from under the limit into the remaining expression.

In this article, you got acquainted with 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 will talk about other types of tasks in future articles, but first you need to learn this lesson in order to move on. We will discuss what to do if there are roots, degrees, we will study infinitesimal equivalent functions, wonderful limits, L'Hopital's rule.

If you can't figure out the limits on your own, don't panic. We are always happy to help!

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 simply infuriates the teacher, I know of cases when part-time students were wrapped 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 instance:

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:

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.

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?

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

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.

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

– 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.

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 instance:

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 A 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 A is a subset of the set V if any X belonging to the set A, belongs to the set V. 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 V. 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:

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 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:

Commutativity.

A È B=BÈ A
AÇ B=BÇ A

Associativity.

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