General definition of the limit of a function. The limit of a sequence and the limit of a function in terms of Cauchy. Derivative of a function defined parametrically

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 a few detailed examples solution 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 but . Dot but 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 more 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 carefully and you will notice that in the numerator we have quadratic equation. 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 definition of the finite limit of a sequence is given. Related properties and an equivalent definition are considered. A definition is given that a point a is not a limit of a sequence. Examples are considered in which the existence of a limit is proved using the definition.

Content

See also: Sequence limit - basic theorems and properties
Main types of inequalities and their properties

Here we consider the definition of the finite limit of a sequence. The case of a sequence converging to infinity is discussed on the page "Definition of an infinitely large sequence".

The limit of a sequence is a number a if for any positive number ε > 0 there is such natural number N ε , depending on ε , such that for all natural numbers n > N ε the inequality
| x n - a|< ε .
Here x n is the element of the sequence with number n . Sequence limit denoted like this:
.
Or at .

Let's transform the inequality:
;
;
.

ε is a neighborhood of the point a is an open interval (a - ε, a + ε ). A convergent sequence is one that has a limit. It is also said that the sequence converges to a. A divergent sequence is a sequence that has no limit.

It follows from the definition that if the sequence has a limit a , then no matter what ε - neighborhood of the point a we choose, it can be outside of it only finite number sequence elements, or none at all (empty set). And any ε - neighborhood contains an infinite number of elements. Indeed, by setting a certain number ε , we thereby have a number . So all elements of the sequence with numbers , by definition, are in the ε - neighborhood of the point a . The first elements can be anywhere. That is, outside the ε - neighborhood there can be no more than elements - that is, a finite number.

We also note that the difference does not have to monotonously tend to zero, that is, to decrease all the time. It can tend to zero not monotonically: it can either increase or decrease, having local maxima. However, these maxima, with increasing n, should tend to zero (perhaps also not monotonously).

Using the logical symbols of existence and universality, the definition of the limit can be written as follows:
(1) .

Determining that a is not a limit

Now consider the converse assertion that the number a is not the limit of the sequence.

Number a is not the limit of the sequence, if there exists such that for any natural n there exists such a natural m >n, what
.

Let's write this statement using logical symbols.
(2) .

The assertion that the number a is not the limit of the sequence, means that
you can choose such an ε - neighborhood of the point a, outside of which there will be an infinite number of elements of the sequence.

Consider an example. Let a sequence with a common element be given
(3)
Any neighborhood of a point contains an infinite number of elements. However, this point is not the limit of the sequence, since any neighborhood of the point also contains an infinite number of elements. Take ε - a neighborhood of a point with ε = 1 . This will be the interval (-1, +1) . All elements except the first one with even n belong to this interval. But all elements with odd n are outside this interval because they satisfy the inequality x n > 2 . Since the number of odd elements is infinite, there will be an infinite number of elements outside the selected neighborhood. Therefore, the point is not the limit of the sequence.

Let us now show this by strictly adhering to assertion (2). The point is not the limit of the sequence (3), because there exists such , so that, for any natural n , there is an odd n for which the inequality
.

It can also be shown that any point a cannot be the limit of this sequence. We can always choose an ε - neighborhood of the point a that does not contain either the point 0 or the point 2. And then there will be an infinite number of elements of the sequence outside the chosen neighborhood.

Equivalent definition of sequence limit

We can give an equivalent definition of the limit of a sequence if we expand the concept of ε - neighborhood. We will get an equivalent definition if instead of ε-neighbourhood, any neighborhood of the point a will appear in it. The neighborhood of a point is any open interval containing that point. Mathematically point neighborhood is defined as follows: , where ε 1 and ε 2 are arbitrary positive numbers.

Then the equivalent definition of the limit is as follows.

The limit of a sequence is such a number a if for any of its neighborhoods there exists such a natural number N , so that all elements of the sequence with numbers belong to this neighborhood.

This definition can also be presented in expanded form.

The limit of a sequence is a number a if for any positive numbers and there exists a natural number N depending on and such that the inequalities hold for all natural numbers
.

Proof of the equivalence of definitions

Let us prove that the above two definitions of the limit of a sequence are equivalent.

Let the number a be the limit of the sequence according to the first definition. This means that there is a function , so that for any positive number ε the following inequalities hold:
(4) at .

Let us show that the number a is the limit of the sequence by the second definition as well. That is, we need to show that there is such a function , so that for any positive numbers ε 1 and ε 2 the following inequalities hold:
(5) at .

Let we have two positive numbers: ε 1 and ε 2 . And let ε be the smallest of them: . Then ; ; . We use this in (5):
.
But the inequalities hold for . Then inequalities (5) also hold for .

That is, we have found a function such that inequalities (5) hold for any positive numbers ε 1 and ε 2 .
The first part is proven.

Now let the number a be the limit of the sequence according to the second definition. This means that there is a function , so that for any positive numbers ε 1 and ε 2 the following inequalities hold:
(5) at .

Let us show that the number a is the limit of the sequence and by the first definition. For this you need to put . Then, for , the following inequalities hold:
.
This corresponds to the first definition with .
The equivalence of the definitions is proved.

Examples

Example 1

Prove that .

(1) .
In our case ;
.

.
Let's use the properties of inequalities. Then if and , then
.

.
Then
at .
This means that the number is the limit of the given sequence:
.

Example 2

Using the definition of the limit of a sequence, prove that
.

We write down the definition of the limit of a sequence:
(1) .
In our case , ;
.

We enter positive numbers and:
.
Let's use the properties of inequalities. Then if and , then
.

That is, for any positive , we can take any natural number greater than or equal to :
.
Then
at .
.

Example 3

We introduce the notation , .
Let's transform the difference:
.
For natural n = 1, 2, 3, ... we have:
.

We write down the definition of the limit of a sequence:
(1) .
We enter positive numbers and:
.
Then if and , then
.

That is, for any positive , we can take any natural number greater than or equal to :
.
Wherein
at .
This means that the number is the limit of the sequence:
.

Example 4

Using the definition of the limit of a sequence, prove that
.

We write down the definition of the limit of a sequence:
(1) .
In our case , ;
.

We enter positive numbers and:
.
Then if and , then
.

That is, for any positive , we can take any natural number greater than or equal to :
.
Then
at .
This means that the number is the limit of the sequence:
.

References:
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.

First definition of the limit of a function (according to Heine)

(x) at point x 0 :
,
if
1) there is such a punctured neighborhood of the point x 0
2) for any sequence ( x n ), converging to x 0 :
, whose elements belong to the neighborhood ,
subsequence (f(xn)) converges to a :
.

Here x 0 and a can be either finite numbers or points at infinity. The neighborhood can be either two-sided or one-sided.

The second definition of the limit of a function (according to Cauchy)

The number a is called the limit of the function f (x) at point x 0 :
,
if
1) there is such a punctured neighborhood of the point x 0 on which the function is defined;
2) for any positive number ε > 0 there exists a number δ ε > 0 , depending on ε, that for all x belonging to a punctured δ ε neighborhood of the point x 0 :
,
function values f (x) belong to ε - neighborhoods of the point a :
.

points x 0 and a can be either finite numbers or points at infinity. The neighborhood can also be both two-sided and one-sided.

We write this definition using the logical symbols of existence and universality:
.

This definition uses neighborhoods with equidistant ends. An equivalent definition can also be given using arbitrary neighborhoods of points.

Definition using arbitrary neighborhoods
The number a is called the limit of the function f (x) at point x 0 :
,
if
1) there is such a punctured neighborhood of the point x 0 on which the function is defined;
2) for any neighborhood U (a) point a there is such a punctured neighborhood of the point x 0 , that for all x that belong to a punctured neighborhood of the point x 0 :
,
function values f (x) belong to the neighborhood U (a) points a :
.

Using the logical symbols of existence and universality, this definition can be written as follows:
.

Unilateral and bilateral limits

The above definitions are universal in the sense that they can be used for any type of neighborhood. If, as we use the left-handed punctured neighborhood of the end point, then we get the definition of the left-handed limit . If we use the neighborhood of a point at infinity as a neighborhood, then we get the definition of the limit at infinity.

To determine the limit according to Heine, this boils down to the fact that an additional restriction is imposed on an arbitrary sequence converging to , that its elements must belong to the corresponding punctured neighborhood of the point .

To determine the Cauchy limit, it is necessary in each case to transform the expressions and into inequalities, using the corresponding definitions of a neighborhood of a point.
See "Neighbourhood of a point".

Determining that a point a is not the limit of a function

Often there is a need to use the condition that the point a is not the limit of the function for . Let us construct negations to the above definitions. In them, we assume that the function f (x) is defined on some punctured neighborhood of the point x 0 . Points a and x 0 can be both finite numbers and infinitely distant. Everything stated below applies to both bilateral and one-sided limits.

According to Heine.
Number a is not limit of the function f (x) at point x 0 : ,
if there is such a sequence ( x n ), converging to x 0 :
,
whose elements belong to the neighborhood ,
what sequence (f(xn)) does not converge to a :
.
.

According to Cauchy.
Number a is not limit of the function f (x) at point x 0 :
,
if there is such a positive number ε > 0 , so that for any positive number δ > 0 , there exists x that belongs to a punctured δ neighborhood of the point x 0 :
,
that the value of the function f (x) does not belong to the ε neighborhood of the point a :
.
.

Of course, if the point a is not the limit of the function at , then this does not mean that it cannot have a limit. Perhaps there is a limit, but it is not equal to a . It is also possible that the function is defined in a punctured neighborhood of the point , but has no limit at .

Function f(x) = sin(1/x) has no limit as x → 0.

For example, the function is defined at , but there is no limit. For proof, we take the sequence . It converges to a point 0 : . Because , then .
Let's take a sequence. It also converges to the point 0 : . But since , then .
Then the limit cannot equal any number a . Indeed, for , there is a sequence with which . Therefore, any non-zero number is not a limit. But it is also not a limit, since there is a sequence with which .

Equivalence of the definitions of the limit according to Heine and according to Cauchy

Theorem
The Heine and Cauchy definitions of the limit of a function are equivalent.

Proof

In the proof, we assume that the function is defined in some punctured neighborhood of the point (finite or at infinity). The point a can also be finite or at infinity.

Heine proof ⇒ Cauchy

Let a function have a limit a at a point according to the first definition (according to Heine). That is, for any sequence that belongs to a punctured neighborhood of the point and has a limit
(1) ,
the limit of the sequence is a :
(2) .

Let us show that the function has a Cauchy limit at a point. That is, for any there exists that for all.

Let's assume the opposite. Let conditions (1) and (2) be satisfied, but the function has no Cauchy limit. That is, there exists such that for any exists , so that
.

Take , where n is a natural number. Then exists and
.
Thus we have constructed a sequence converging to , but the limit of the sequence is not equal to a . This contradicts the condition of the theorem.

The first part is proven.

Cauchy proof ⇒ Heine

Let a function have a limit a at a point according to the second definition (according to Cauchy). That is, for any there exists that
(3) for all .

Let us show that the function has a limit a at a point according to Heine.
Let's take an arbitrary number. According to Cauchy's definition, there exists a number , so (3) holds.

Take an arbitrary sequence belonging to the punctured neighborhood and converging to . By the definition of a convergent sequence, for any there exists such that
at .
Then from (3) it follows that
at .
Since this holds for any , then
.

The theorem has been proven.

References:
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.

See also:

Today at the lesson we will analyze strict sequencing And strict definition of the limit of a function, as well as learn how to solve the corresponding problems of a theoretical nature. The article is intended primarily for first-year students of natural sciences and engineering specialties who have begun to study the theory of mathematical analysis and have encountered difficulties in understanding this section of higher mathematics. In addition, the material is quite accessible to high school students.

Over the years of the site’s existence, I received an unkind dozen of letters with approximately the following content: “I don’t understand mathematical analysis well, what should I do?”, “I don’t understand matan at all, I’m thinking of quitting my studies,” etc. Indeed, it is the matan who often thins out the student group after the very first session. Why are things like this? Because the subject is unthinkably complex? Not at all! The theory of mathematical analysis is not so difficult as it is peculiar. And you need to accept and love her for who she is =)

Let's start with the most difficult case. First and foremost, don't drop out of school. Understand correctly, quit, it will always have time ;-) Of course, if in a year or two from the chosen specialty it will make you sick, then yes - you should think about it (and not smack the fever!) about changing activities. But for now it's worth continuing. And, please, forget the phrase “I don’t understand anything” - it doesn’t happen that you don’t understand anything at all.

What to do if the theory is bad? By the way, this applies not only to mathematical analysis. If the theory is bad, then first you need to SERIOUSLY put on practice. At the same time, two strategic tasks are solved at once:

- First, a significant proportion theoretical knowledge came about through practice. And so many people understand theory through ... - that's right! No, no, you didn't think about that.

- And, secondly, practical skills are very likely to “stretch” you in the exam, even if ..., but let's not tune in like that! Everything is real and everything is really “lifted” in a fairly short time. Mathematical analysis is my favorite section of higher mathematics, and therefore I simply could not help but lend you a helping hand:

At the beginning of the 1st semester, sequence limits and function limits usually pass. Do not understand what it is and do not know how to solve them? Start with an article Function Limits, in which the concept itself is considered “on the fingers” and the simplest examples are analyzed. Then work through other lessons on the topic, including a lesson about within sequences, on which I have actually already formulated a rigorous definition.

What icons besides inequality signs and modulus do you know?

- a long vertical stick reads like this: “such that”, “such that”, “such that” or “such that”, in our case, obviously, we are talking about a number - therefore “such that”;

- for all "en" greater than ;

– module sign means distance, i.e. this entry tells us that the distance between values is less than epsilon.

Well, is it deadly difficult? =)

After mastering the practice, I am waiting for you in the following paragraph:

Indeed, let's think a little - how to formulate a rigorous definition of a sequence? ... The first thing that comes to mind in the light practical session: "the limit of a sequence is the number to which the members of the sequence approach infinitely close."

Okay, let's write subsequence :

It is easy to grasp that subsequence approach infinitely close to -1, and even-numbered terms - to "unit".

Maybe two limits? But then why can't some sequence have ten or twenty of them? That way you can go far. In this regard, it is logical to assume that if the sequence has a limit, then it is unique.

Note : the sequence has no limit, but two subsequences can be distinguished from it (see above), each of which has its own limit.

Thus, the above definition turns out to be untenable. Yes, it works for cases like (which I did not quite correctly use in simplified explanations of practical examples), but now we need to find a strict definition.

Attempt two: “the limit of a sequence is the number that ALL members of the sequence approach, with the exception of, perhaps, their final quantities." This is closer to the truth, but still not entirely accurate. So, for example, the sequence half of the terms do not approach zero at all - they are simply equal to it =) By the way, the "flashing light" generally takes two fixed values.

The formulation is not difficult to clarify, but then another question arises: how to write the definition in mathematical signs? scientific world struggled with this problem for a long time, until the situation was resolved famous maestro, which, in essence, formalized the classical mathematical analysis in all its rigor. Cauchy offered to operate surroundings which greatly advanced the theory.

Consider some point and its arbitrary-neighborhood:

The value of "epsilon" is always positive, and moreover, we have the right to choose it ourselves. Assume that the given neighborhood contains a set of terms (not necessarily all) some sequence. How to write down the fact that, for example, the tenth term fell into the neighborhood? Let it be on the right side of it. Then the distance between the points and should be less than "epsilon": . However, if the "x tenth" is located to the left of the point "a", then the difference will be negative, and therefore the sign must be added to it module: .

Definition: a number is called the limit of a sequence if for any its surroundings (preselected) there is a natural number - SUCH that ALL members of the sequence with higher numbers will be inside the neighborhood:

Or shorter: if

In other words, no matter how small the value of "epsilon" we take, sooner or later the "infinite tail" of the sequence will FULLY be in this neighborhood.

So, for example, the "infinite tail" of the sequence FULLY goes into any arbitrarily small -neighborhood of the point . Thus, this value is the limit of the sequence by definition. I remind you that a sequence whose limit is zero is called infinitesimal.

It should be noted that for the sequence it is no longer possible to say "infinite tail will come”- members with odd numbers are in fact equal to zero and “do not go anywhere” \u003d) That is why the verb “will end up” is used in the definition. And, of course, the members of such a sequence as also "do not go anywhere." By the way, check if the number will be its limit.

Let us now show that the sequence has no limit. Consider, for example, a neighborhood of the point . It is quite clear that there is no such number, after which ALL members will be in the given neighborhood - odd members will always "jump" to "minus one". For a similar reason, there is no limit at the point .

Fix the material with practice:

Example 1

Prove that the limit of the sequence is zero. Specify the number after which all members of the sequence are guaranteed to be inside any arbitrarily small -neighborhood of the point .

Note : for many sequences, the desired natural number depends on the value - hence the notation .

Solution: consider arbitrary will there be number - such that ALL members with higher numbers will be inside this neighborhood:

To show the existence of the required number , we express in terms of .

Since for any value "en", then the modulus sign can be removed:

We use "school" actions with inequalities that I repeated in the lessons Linear inequalities And Function scope. In this case, an important circumstance is that "epsilon" and "en" are positive:

Since on the left we are talking about natural numbers, and the right side in general case fractional, then it must be rounded:

Note : sometimes a unit is added to the right for reinsurance, but in fact this is an overkill. Relatively speaking, if we also weaken the result by rounding down, then the nearest suitable number (“three”) will still satisfy the original inequality.

And now we look at inequality and remember that initially we considered arbitrary-neighborhood, i.e. "epsilon" can be equal to anyone positive number.

Output: for any arbitrarily small -neighborhood of the point, the value . Thus, a number is the limit of a sequence by definition. Q.E.D.

By the way, from the result a natural pattern is clearly visible: the smaller the -neighborhood, the greater the number after which ALL members of the sequence will be in this neighborhood. But no matter how small the "epsilon" is, there will always be an "infinite tail" inside, and outside - even if it is large, however final number of members.

How are the impressions? =) I agree that it is strange. But strictly! Please re-read and think again.

Consider a similar example and get acquainted with others techniques:

Example 2

Solution: by the definition of a sequence, it is necessary to prove that (Speak out loud!!!).

Consider arbitrary-neighborhood of the point and check, does it exist natural number - such that for all larger numbers the following inequality holds:

To show the existence of such an , you need to express "en" through "epsilon". We simplify the expression under the module sign:

The module destroys the minus sign:

The denominator is positive for any "en", therefore, the sticks can be removed:

Shuffling:

Now we need to extract Square root, but the catch is that for some epsilons, the right hand side will be negative. To avoid this trouble let's strengthen inequality modulus:

Why can this be done? If, relatively speaking, it turns out that , then the condition will be satisfied even more so. The module can just increase wanted number , and that will suit us too! Roughly speaking, if the hundredth is suitable, then the two hundredth will do! According to the definition, you need to show the very existence of the number(at least some), after which all members of the sequence will be in -neighbourhood. By the way, that is why we are not afraid of the final rounding of the right side up.

Extracting the root:

And round the result:

Output: because the value of "epsilon" was chosen arbitrarily, then for any arbitrarily small -neighborhood of the point, the value , such that the inequality . In this way, by definition. Q.E.D.

I advise especially understand the strengthening and weakening of inequalities - these are typical and very common methods of mathematical analysis. The only thing you need to monitor the correctness of this or that action. So, for example, the inequality by no means loosen, subtracting, say, one:

Again, conditional: if the number fits exactly, then the previous one may no longer fit.

The following example is for a standalone solution:

Example 3

Using the definition of a sequence, prove that

Short solution and answer at the end of the lesson.

If the sequence infinitely great, then the definition of the limit is formulated in a similar way: a point is called the limit of a sequence if for any, arbitrarily large there is a number such that for all larger numbers , the inequality will be satisfied. The number is called the neighborhood of the point "plus infinity":

In other words, whatever great importance no matter what, the “infinite tail” of the sequence will necessarily go into the -neighborhood of the point , leaving only a finite number of terms on the left.

Working example:

And an abbreviated notation: if

For the case, write the definition yourself. The correct version is at the end of the lesson.

After you have "filled" your hand with practical examples and figured out the definition of the limit of a sequence, you can turn to the literature on mathematical analysis and / or your lecture book. I recommend downloading the 1st volume of Bohan (easier - for part-time students) and Fikhtengoltz (more detailed and thorough). Of the other authors, I advise Piskunov, whose course is focused on technical universities.

Try to conscientiously study the theorems that concern the limit of the sequence, their proofs, consequences. At first, the theory may seem "cloudy", but this is normal - it just takes some getting used to. And many will even get a taste!

Strict definition of the limit of a function

Let's start with the same thing - how to formulate this concept? The verbal definition of the limit of a function is formulated much more simply: “a number is the limit of a function, if with“ x ” tending to (both left and right), the corresponding values of the function tend to » (see drawing). Everything seems to be normal, but words are words, meaning is meaning, an icon is an icon, and strict mathematical notation is not enough. And in the second paragraph, we will get acquainted with two approaches to solving this issue.

Let the function be defined on some interval except, possibly, for the point . IN educational literature it is generally accepted that the function is there not defined:

This choice highlights the essence of the function limit: "x" infinitely close approaches , and the corresponding values of the function are infinitely close to . In other words, the concept of a limit does not imply an “exact approach” to points, namely infinitely close approximation, it does not matter whether the function is defined at the point or not.

The first definition of the limit of a function, not surprisingly, is formulated using two sequences. Firstly, the concepts are related, and secondly, the limits of functions are usually studied after the limits of sequences.

Consider the sequence points (not on the drawing), belonging to the interval And other than, which converges to . Then the corresponding values of the function also form a numerical sequence, the members of which are located on the y-axis.

Heine function limit for any point sequences (belonging to and different from), which converges to the point , the corresponding sequence of function values converges to .

Eduard Heine is a German mathematician. ... And there is no need to think anything like that, there is only one gay in Europe - this is Gay-Lussac =)

The second definition of the limit was built ... yes, yes, you are right. But first, let's look at its design. Consider an arbitrary -neighbourhood of the point ("black" neighborhood). Based on the previous paragraph, the notation means that some value function is located inside the "epsilon"-environment.

Now let's find -neighborhood that corresponds to the given -neighborhood (mentally draw black dotted lines from left to right and then from top to bottom). Note that the value is chosen along the length of the smaller segment, in this case, along the length of the shorter left segment. Moreover, the "crimson" -neighborhood of a point can even be reduced, since in the following definition the very fact of existence is important this neighbourhood. And, similarly, the entry means that some value is inside the "delta" neighborhood.

Cauchy limit of a function: the number is called the limit of the function at the point if for any preselected neighborhood (arbitrarily small), exists-neighborhood of the point , SUCH that: AS ONLY values (owned) included in this area: (red arrows)- SO IMMEDIATELY the corresponding values of the function are guaranteed to enter the -neighborhood: (blue arrows).

I must warn you that in order to be more intelligible, I improvised a little, so do not abuse it =)

Shorthand: if

What is the essence of the definition? Figuratively speaking, by infinitely decreasing the -neighbourhood, we "accompany" the values of the function to its limit, leaving no alternative for them to approach somewhere else. Quite unusual, but again strictly! To get the idea right, reread the wording again.

! Attention: if you need to formulate only definition according to Heine or only Cauchy definition please don't forget about significant preliminary comment: "Consider a function that is defined on some interval except perhaps a point". I stated this once at the very beginning and did not repeat it each time.

According to the corresponding theorem of mathematical analysis, the Heine and Cauchy definitions are equivalent, but the second variant is the most well-known (still would!), which is also called the "limit on the tongue":

Example 4

Using the definition of a limit, prove that

Solution: the function is defined on the entire number line except for the point . Using the definition of , we prove the existence of a limit at a given point.

Note : the magnitude of the "delta" neighborhood depends on the "epsilon", hence the designation

Consider arbitrary-neighborhood. The task is to use this value to check whether does it exist- neighborhood, SUCH, which from the inequality follows the inequality .

Assuming that , we transform the last inequality:
(decompose the square trinomial)

Definition 1. Let E- an infinite number. If any neighborhood contains points of the set E, different from the point but, then but called marginal set point E.

Definition 2. (Heinrich Heine (1821-1881)). Let the function
defined on the set X And BUT called limit functions
at the point (or when
, if for any sequence of argument values
, converging to , the corresponding sequence of function values converges to the number BUT. Write:
.

Examples. 1) Function
has a limit equal to from, at any point on the number line.

Indeed, for any point and any sequence of argument values
, converging to and consisting of numbers other than , the corresponding sequence of function values has the form
, and we know that this sequence converges to from. That's why
.

2) For function

.

This is obvious, because if
, then and
.

3) Dirichlet function
has no limit at any point.

Indeed, let
And
, and all are rational numbers. Then
for all n, that's why
. If
and all are irrational numbers, then
for all n, that's why
. We see that the conditions of Definition 2 are not satisfied, therefore
does not exist.

4)
.

Indeed, take an arbitrary sequence
, converging to

number 2. Then . Q.E.D.

Definition 3. (Cauchy (1789-1857)). Let the function
defined on the set X And is the limit point of this set. Number BUT called limit functions
at the point (or when
, if for any
there will be
, such that for all values of the argument X satisfying the inequality

the inequality

Write:
.

The definition of Cauchy can also be given with the help of neighborhoods, if you notice that , a:

let the function
defined on the set X And is the limit point of this set. Number BUT called the limit functions
at the point , if for any -neighborhood of a point BUT
there is a pierced - neighborhood of the point
, such that
.

It is useful to illustrate this definition with a figure.

Example 5.
.

Indeed, let's take
arbitrarily and find
, such that for all X satisfying the inequality
the inequality
. The last inequality is equivalent to the inequality
, so we see that it suffices to take
. The assertion has been proven.

Fair

Theorem 1. The definitions of the limit of a function according to Heine and according to Cauchy are equivalent.

Proof. 1) Let
by Cauchy. Let us prove that the same number is also the limit according to Heine.

Let's take
arbitrarily. According to Definition 3, there exists
, such that for all
the inequality
. Let be
is an arbitrary sequence such that
at
. Then there is a number N such that for everyone
the inequality
, that's why
for all
, i.e.

according to Heine.

2) Let now
according to Heine. Let's prove that
and according to Cauchy.

Assume the opposite, i.e. what
by Cauchy. Then there is
such that for any
there will be
,
And
. Consider the sequence
. For the specified
and any n exists

And
. It means that
, although
, i.e. number BUT is not the limit
at the point according to Heine. We have obtained a contradiction, which proves the assertion. The theorem has been proven.

Theorem 2 (on the uniqueness of the limit). If there is a limit of a function at a point , then it is the only one.

Proof. If the limit is defined in the sense of Heine, then its uniqueness follows from the uniqueness of the limit of the sequence. If the limit is defined according to Cauchy, then its uniqueness follows from the equivalence of the definitions of the limit according to Cauchy and according to Heine. The theorem has been proven.

Similarly to the Cauchy criterion for sequences, there is a Cauchy criterion for the existence of a limit of a function. Before formulating it, we give

Definition 4. They say that the function
satisfies the Cauchy condition at the point , if for any
exists

, such that
And
, the inequality
.

Theorem 3 (Cauchy's criterion for the existence of a limit). In order for the function
had at the point finite limit, it is necessary and sufficient that at this point the function satisfies the Cauchy condition.

Proof.Need. Let be
. We have to prove that
satisfies at the point the Cauchy condition.

Let's take
arbitrarily and put
. By definition of the limit for exists
, such that for any values
satisfying the inequalities
And
, the inequalities
And
. Then

The need has been proven.

Adequacy. Let the function
satisfies at the point the Cauchy condition. Gotta prove that she has a point end limit.

Let's take
arbitrarily. By Definition 4, there is
, such that from the inequalities
,
follows that
- it is given.

Let us first show that for any sequence
, converging to , subsequence
function values converge. Indeed, if
, then, by virtue of the definition of the limit of the sequence, for a given
there is a number N, such that for any

And
. Insofar as
at the point satisfies the Cauchy condition, we have
. Then, by the Cauchy criterion for sequences, the sequence
converges. Let us show that all such sequences
converge to the same limit. Assume the opposite, i.e. what are sequences
And
,
,
, such that. Let's consider a sequence. It is clear that it converges to , therefore, by the above, the sequence converges, which is impossible, since the subsequences
And
have different limits And . The obtained contradiction shows that =. Therefore, by Heine's definition, a function has at a point end limit. The sufficiency, and hence the theorem, are proved.