Function limit. Universal definition of the limit of a function according to Hein and Cauchy

(x) at point x 0 :
,
If
1) there is such a punctured neighborhood of the point x 0
2) for any sequence (xn), 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.

.

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 is such a number δ ε > 0 , depending on ε, that for all x belonging to the punctured δ ε - neighborhood of the point x 0 :
,
function values ​​f (x) belong to the ε-neighborhood of point a:
.

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

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

This definition uses neighborhoods with equidistant ends. An equivalent definition can 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) of point a there is such a punctured neighborhood of point x 0 that for all x belonging to the 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:
.

One-sided and two-sided 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-sided punctured neighborhood end point, then we obtain the definition of a left-sided limit. If we use the neighborhood of a point at infinity as a neighborhood, we obtain the definition of the limit at infinity.

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

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

Determining that point a is not the limit of a function

It often becomes necessary to use the condition that point a is not the limit of the function at . 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 either finite numbers or infinitely distant. Everything stated below applies to both bilateral and unilateral limits.

According to Heine.
Number a is not limit of the function f (x) at point x 0 : ,
if such a sequence exists (xn), converging to x 0 :
,
whose elements belong to the neighborhood,
what is the 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 for any positive number δ > 0 , there exists an x ​​that belongs to the punctured δ-neighborhood of the point x 0 :
,
that the value of the function f (x) does not belong to the ε-neighborhood of point a:
.
.

Of course, if point a is not the limit of a function at , this does not mean that it cannot have a limit. There may be 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, a function is defined at , but there is no limit. To prove it, let's take the sequence . It converges to a point 0 : . Because , then .
Let's take the sequence. It also converges to the point 0 : . But since , then .
Then the limit cannot be equal to 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 Heine and Cauchy definitions of the limit

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 a point (finite or at infinity). Point a can also be finite or at infinity.

Heine's proof ⇒ Cauchy's

Let the function have a limit a at a point according to the first definition (according to Heine). That is, for any sequence belonging to a neighborhood of a point and having 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 everyone there is something that is for everyone.

Let's assume the opposite. Let conditions (1) and (2) be satisfied, but the function does not have a Cauchy limit. That is, there is something that exists for anyone, so
.

Let's take , where n - natural number. Then there exists , and
.
Thus we have constructed a sequence converging to , but the limit of the sequence is not equal to a . This contradicts the conditions of the theorem.

The first part has been proven.

Cauchy's proof ⇒ Heine's

Let the function have a limit a at a point according to the second definition (according to Cauchy). That is, for anyone there is 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, the number exists, so (3) holds.

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

The theorem has been proven.

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

Here we will look at 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”.

Definition .
(xn), if for any positive number ε > 0 there is a natural number N ε depending on ε such that for all natural numbers n > N ε the inequality
| x n - a|< ε .
The sequence limit is denoted as follows:
.
Or at .

Let's transform the inequality:
;
;
.

An open interval (a - ε, a + ε) is called ε - neighborhood of point a.

A sequence that has a limit is called convergent sequence. It is also said that the sequence converges to a. A sequence that has no limit is called divergent.

From the definition it follows that if a sequence has a limit a, that no matter what ε-neighborhood of point a we choose, beyond its limits there can be only a finite number of elements of the sequence, or none at all ( empty set). And any ε-neighborhood contains an infinite number of elements. In fact, having given a certain number ε, we thereby have the number . So all elements of the sequence with numbers , by definition, are located in the ε - neighborhood of point a . The first elements can be located 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 monotonically tend to zero, that is, decrease all the time. It can tend to zero non-monotonically: it can either increase or decrease, having local maxima. However, these maxima, as n increases, should tend to zero (possibly also not monotonically).

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

Determining that a is not a limit

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

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

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

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

Let's look at 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. Let's 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, since 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 chosen neighborhood. Therefore, the point is not the limit of the sequence.

Now we will show this, strictly adhering to statement (2). The point is not a limit of the sequence (3), since there exists such that, for any natural n, there is an odd one for which the inequality holds
.

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

Equivalent definition

We can give an equivalent definition of the limit of a sequence if we expand the concept of ε - neighborhood. We will obtain an equivalent definition if, instead of an ε-neighborhood, it contains any neighborhood of the point a.

Determining the neighborhood of a point
Neighborhood of point a any open interval containing this point is called. Mathematically, the neighborhood is defined as follows: , where ε 1 and ε 2 - arbitrary positive numbers.

Then the definition of the limit will be as follows.

Equivalent definition of sequence limit
The number a is called the limit of the sequence, if for any neighborhood of it there is a natural number N such that all elements of the sequence with numbers belong to this neighborhood.

This definition can also be presented in expanded form.

The number a is called the limit of the sequence, 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 equivalence of definitions

Let us prove that the two definitions of the limit of a sequence presented above 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 are satisfied:
(4) at .

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

Let us have two positive numbers: ε 1 and ε 2 . And let ε be the smallest of them: . Then ; ; . Let's use this in (5):
.
But the inequalities are satisfied for . Then inequalities (5) are also satisfied for .

That is, we have found a function for which inequalities (5) are satisfied for any positive numbers ε 1 and ε 2 .
The first part has been proven.

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

Let us show that the number a is the limit of the sequence by the first definition. To do this you need to put . Then when the following inequalities hold:
.
This corresponds to the first definition with .
The equivalence of the definitions has been proven.

Examples

Here we will look at several examples in which we need to prove that a given number a is the limit of a sequence. In this case, you need to specify an arbitrary positive number ε and define a function N of ε such that the inequality .

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
.

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

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:
.

Let us write down the definition of the limit of a sequence:
(1) .
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
.

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

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.

Today in class we will look at strict sequencing And strict definition of the limit of a function, and also learn to solve relevant problems theoretical in nature. The article is intended primarily for first-year students of natural sciences and engineering specialties who began to study the theory of mathematical analysis and 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 have received a dozen letters with approximately the following content: “I don’t understand mathematical analysis well, what should I do?”, “I don’t understand math at all, I’m thinking of quitting my studies,” etc. And indeed, it is the matan who often thins out the student group after the first session. Why is this the case? Because the subject is unimaginably 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. The first and most important thing is that you don’t have to give up your studies. Understand correctly, you can always quit;-) Of course, if after a year or two you feel sick from your chosen specialty, then yes, you should think about it (and don't get mad!) about a change of activity. 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? This, by the way, applies not only to mathematical analysis. If the theory is bad, then first you need to SERIOUSLY focus on practice. In this case, two strategic tasks are solved at once:

– Firstly, a significant share of theoretical knowledge emerged through practice. And that’s why many people understand the theory through... – that’s right! No, no, you're not thinking about that =)

– And, secondly, practical skills will most likely “pull” you through the exam, even if... but let’s not get so excited! Everything is real and everything can be “raised” in a fairly short time. Mathematical analysis is my favorite section of higher mathematics, and therefore I simply could not help but give you a helping hand:

At the beginning of the 1st semester, sequence limits and function limits are usually covered. Don’t understand what these are and don’t know how to solve them? Start with the article Function limits, in which the concept itself is examined “on the fingers” and the simplest examples are analyzed. Next, work through other lessons on the topic, including a lesson about within sequences, on which I have actually already formulated a strict definition.

What symbols 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 ;

– the modulus 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 look forward to seeing you in the next paragraph:

And in fact, let's think a little - how to formulate a strict definition of sequence? ...The first thing that comes to mind in the world practical lesson: “the limit of a sequence is the number to which the members of the sequence approach infinitely close.”

Okay, let's write it down subsequence :

It is not difficult to understand that subsequence approach infinitely close to the number –1, and even-numbered terms – to “one”.

Or maybe there are two limits? But then why can’t any sequence have ten or twenty of them? You can go far this way. In this regard, it is logical to assume that if a 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 use quite correctly 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 to which ALL members of the sequence approach, except 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 I struggled with this problem for a long time until I resolved the situation famous maestro, which, in essence, formalized classical mathematical analysis in all its rigor. Cauchy suggested surgery surroundings , which significantly advanced the theory.

Consider some point and its arbitrary-surroundings:

The value of "epsilon" is always positive, and, moreover, we have the right to choose it ourselves. Let us assume that in this neighborhood there are many members (not necessarily all) some sequence. How to write down the fact that, for example, the tenth term is in 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 “x tenth” is located to the left of 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 (pre-selected) there is a natural number SUCH that ALL members of the sequence with higher numbers will be inside the neighborhood:

Or in short: if

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

For example, the “infinite tail” of the sequence will COMPLETELY enter any arbitrarily small neighborhood of the point . So this value is the limit of the sequence by definition. Let me remind you that the sequence whose limit equal to zero, called infinitesimal.

It should be noted that for a sequence it is no longer possible to say “endless tail” will come in» – members with odd numbers in fact they are equal to zero and “they don’t go anywhere” =) That is why the verb “will appear” is used in the definition. And, of course, the members of a sequence like this also “go nowhere.” By the way, check whether the number is its limit.

Now we will show that the sequence has no limit. Consider, for example, a neighborhood of the point . It is absolutely clear that there is no such number after which ALL terms will end up in a given neighborhood - odd terms will always “jump out” to “minus one”. For a similar reason, there is no limit at the point.

Let's consolidate 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 required natural number depends on the value - hence the notation .

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

To show the existence of the required number, we express it through .

Since for any value of “en”, the modulus sign can be removed:

We use “school” actions with inequalities that I repeated in class Linear inequalities And Function Domain. In this case, an important circumstance is that “epsilon” and “en” are positive:

Since we are talking about natural numbers on the left, and the right side is generally fractional, it needs to be rounded:

Note : sometimes a unit is added to the right to be on the safe side, but in reality this is overkill. Relatively speaking, if we weaken the result by rounding down, then the nearest suitable number (“three”) will still satisfy the original inequality.

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

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

By the way, from the result obtained a natural pattern is clearly visible: the smaller the neighborhood, the larger 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 your impressions? =) I agree that it’s a bit strange. But strictly! Please re-read and think about everything again.

Let's look at a similar example and get acquainted with other technical techniques:

Example 2

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

Let's 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 , you need to express “en” through “epsilon”. We simplify the expression under the modulus sign:

The module destroys the minus sign:

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

Shuffle:

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

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

Extracting the root:

And round the result:

Conclusion: because the value “epsilon” was chosen arbitrarily, then for any arbitrarily small neighborhood of the point the value was found , such that for all larger numbers the inequality holds . Thus, a-priory. Q.E.D.

I advise especially understanding the strengthening and weakening of inequalities is a typical and very common technique in mathematical analysis. The only thing you need to monitor is the correctness of this or that action. So, for example, inequality under no circumstances is it possible loosen, subtracting, say, one:

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

The following example is for independent decision:

Example 3

Using the definition of a sequence, prove that

Quick Solution and the answer at the end of the lesson.

If the sequence infinitely large, then the definition of the limit is formulated in a similar way: a point is called the limit of a sequence if for any, as big as you like number, there is a number such that for all larger numbers, the inequality will be satisfied. The number is called vicinity of the point “plus infinity”:

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

Standard example:

And shortened notation: , if

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

Once you've gotten your head around practical examples and figured out the definition of the limit of a sequence, you can turn to the literature on calculus and/or your lecture notebook. I recommend downloading volume 1 of Bohan (simpler - for correspondence students) and Fichtenholtz (in more detail and detail). Among other authors, I recommend Piskunov, whose course is aimed at 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 - you just need to get used to it. And many will even get a taste for it!

Rigorous 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 simpler: “a number is the limit of a function if with “x” tending to (both left and right), the corresponding function values ​​tend to » (see drawing). Everything seems to be normal, but words are words, meaning is meaning, an icon is an icon, but strict mathematical notation not enough. And in the second paragraph we will get acquainted with two approaches to solving this issue.

Let the function be defined on a certain interval, with the possible exception of the point. IN educational literature it is generally accepted that the function is there Not defined:

This choice emphasizes the essence of the limit of a function: "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 “exact approach” to points, but 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, limits of functions are usually studied after limits of sequences.

Consider the sequence points (not on the drawing), belonging to the interval and different from, which converges To . Then the corresponding function values ​​also form a numerical sequence, the members of which are located on the ordinate axis.

Limit of a function according to Heine for any sequences of points (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 - Gay-Lussac =)

The second definition of the limit was created... yes, yes, you are right. But first, let's understand its design. Consider an arbitrary -neighborhood of the point (“black” neighborhood). Based on the previous paragraph, the entry means that some value function is located inside the “epsilon” neighborhood.

Now we find the -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 selected along the length of the smaller segment, in in this case– along the length of the shorter left segment. Moreover, the “raspberry” -neighborhood of a point can even be reduced, since in the following definition the very fact of existence is important this neighborhood. And, similarly, the notation means that some value is within the “delta” neighborhood.

Cauchy function limit: a number is called the limit of a function at a point if for any pre-selected neighborhood (as small as you like), exists-neighborhood of the point, SUCH, that: AS ONLY values (belonging to) included in this area: (red arrows)– SO IMMEDIATELY the corresponding function values ​​are guaranteed to enter the -neighborhood: (blue arrows).

I must warn you that for the sake of clarity, I improvised a little, so do not overuse =)

Short entry: , if

What is the essence of the definition? Figuratively speaking, by infinitely decreasing the -neighborhood, we “accompany” the function values ​​to their limit, leaving them no alternative to approaching somewhere else. Quite unusual, but again strict! To fully understand the idea, re-read the wording again.

! Attention: if you only need to formulate Heine's definition or just Cauchy definition please don't forget about significant preliminary comments: "Consider a function that is defined on a certain interval, with the possible exception of a point". I stated this once at the very beginning and did not repeat it every time.

According to the corresponding theorem of mathematical analysis, the Heine and Cauchy definitions are equivalent, but the second option is the most famous (still would!), which is also called the "language limit":

Example 4

Using the definition of limit, prove that

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

Note : the value of the “delta” neighborhood depends on the “epsilon”, hence the designation

Let's consider arbitrary-surroundings. The task is to use this value to check whether does it exist-surroundings, SUCH, which from the inequality inequality follows .

Assuming that , we transform the last inequality:
(expanded the quadratic trinomial)

Function limit- number a will be the limit of some variable quantity if, in the process of its change, this variable quantity indefinitely approaches a.

Or in other words, the number A is the limit of the function y = f(x) at the point x 0, if for any sequence of points from the domain of definition of the function , not equal x 0, and which converges to the point x 0 (lim x n = x0), the sequence of corresponding function values ​​converges to the number A.

The graph of a function whose limit, given an argument that tends to infinity, is equal to L:

Meaning A is limit (limit value) of the function f(x) at the point x 0 in case for any sequence of points , which converges to x 0, but which does not contain x 0 as one of its elements (i.e. in the punctured vicinity x 0), sequence of function values converges to A.

Limit of a Cauchy function.

Meaning A will be limit of the function f(x) at the point x 0 if for any non-negative number taken in advance ε the corresponding non-negative number will be found δ = δ(ε) such that for each argument x, satisfying the condition 0 < | x - x0 | < δ , the inequality will be satisfied | f(x)A |< ε .

It will be very simple if you understand the essence of the limit and the basic rules for finding it. What is the limit of the function f (x) at x striving for a equals A, is written like this:

Moreover, the value to which the variable tends x, can be not only a number, but also infinity (∞), sometimes +∞ or -∞, or there may be no limit at all.

To understand how find the limits of a function, it is best to look at examples of solutions.

It is necessary to find the limits of the function f (x) = 1/x at:

x→ 2, x→ 0, x→ ∞.

Let's find a solution to the first limit. To do this, you can simply substitute x the number it tends to, i.e. 2, we get:

Let's find the second limit of the function. Substitute here in pure form 0 instead x it is impossible, because You cannot divide by 0. But we can take values ​​close to zero, for example, 0.01; 0.001; 0.0001; 0.00001 and so on, and the value of the function f (x) will increase: 100; 1000; 10000; 100,000 and so on. Thus, it can be understood that when x→ 0 the value of the function that is under the limit sign will increase without limit, i.e. strive towards infinity. Which means:

Regarding the third limit. The same situation as in the previous case, it is impossible to substitute ∞ in its purest form. We need to consider the case of unlimited increase x. We substitute 1000 one by one; 10000; 100000 and so on, we have that the value of the function f (x) = 1/x will decrease: 0.001; 0.0001; 0.00001; and so on, tending to zero. That's why:

It is necessary to calculate the limit of the function

Starting to solve the second example, we see uncertainty. From here we find the highest degree of the numerator and denominator - this is x 3, we take it out of brackets in the numerator and denominator and then reduce it by:

Answer

The first step in finding this limit, substitute the value 1 instead x, resulting in uncertainty. To solve it, let’s factorize the numerator and do this using the method of finding roots quadratic equation x 2 + 2x - 3:

D = 2 2 - 4*1*(-3) = 4 +12 = 16→ √ D=√16 = 4

x 1.2 = (-2±4)/2→ x 1 = -3;x 2= 1.

So the numerator will be:

Answer

This is the definition of it specific meaning or a specific area where a function that is limited by a limit falls.

To solve limits, follow the rules:

Having understood the essence and main rules for solving the limit, You'll get basic concept about how to solve them.

Mathematics is the science that builds the world. Both the scientist and the common man - no one can do without it. First, young children are taught to count, then to add, subtract, multiply and divide, to high school come into play letter designations, and in the older age you can’t do without them.

But today we will talk about what all known mathematics is based on. About a community of numbers called “sequence limits”.

What are sequences and where is their limit?

The meaning of the word “sequence” is not difficult to interpret. This is an arrangement of things where someone or something is located in a certain order or queue. For example, the queue for tickets to the zoo is a sequence. And there can only be one! If, for example, you look at the queue at the store, this is one sequence. And if one person from this queue suddenly leaves, then this is a different queue, a different order.

The word “limit” is also easily interpreted - it is the end of something. However, in mathematics, the limits of sequences are those values ​​on the number line to which a sequence of numbers tends. Why does it strive and not end? It's simple, the number line has no end, and most sequences, like rays, have only a beginning and look like this:

x 1, x 2, x 3,...x n...

Hence the definition of a sequence is a function of the natural argument. More in simple words is a series of members of a certain set.

How is the number sequence constructed?

The simplest example number sequence might look like this: 1, 2, 3, 4, …n…

In most cases, for practical purposes, sequences are built from numbers, and each next member of the series, let's denote it X, has its own name. For example:

x 1 is the first member of the sequence;

x 2 is the second term of the sequence;

x 3 is the third term;

x n is the nth term.

IN practical methods the sequence is given by a general formula in which there is some variable. For example:

X n =3n, then the series of numbers itself will look like this:

It is worth remembering that when recording sequences in general, you can use any letters, and not just X. For example: y, z, k, etc.

Arithmetic progression as part of sequences

Before looking for the limits of sequences, it is advisable to plunge deeper into the very concept of such a number series, which everyone encountered when they were in middle school. An arithmetic progression is a series of numbers in which the difference between adjacent terms is constant.

Problem: “Let a 1 = 15, and the progression step of the number series d = 4. Construct the first 4 terms of this series"

Solution: a 1 = 15 (by condition) is the first term of the progression (number series).

and 2 = 15+4=19 is the second term of the progression.

and 3 =19+4=23 is the third term.

and 4 =23+4=27 is the fourth term.

However, using this method it is difficult to reach large values, for example up to a 125. . Especially for such cases, a formula convenient for practice was derived: a n =a 1 +d(n-1). In this case, a 125 =15+4(125-1)=511.

Types of sequences

Most of the sequences are endless, it's worth remembering for the rest of your life. There are two interesting looking number series. The first is given by the formula a n =(-1) n. Mathematicians often call this sequence a flasher. Why? Let's check its number series.

1, 1, -1, 1, -1, 1, etc. On similar example it becomes clear that numbers in sequences can easily be repeated.

Factorial sequence. It's easy to guess - the formula defining the sequence contains a factorial. For example: a n = (n+1)!

Then the sequence will look like this:

a 2 = 1x2x3 = 6;

and 3 = 1x2x3x4 = 24, etc.

Sequence given arithmetic progression, is called infinitely decreasing if the inequality -1 is observed for all its terms

and 3 = - 1/8, etc.

There is even a sequence consisting of the same number. So, n =6 consists of an infinite number of sixes.

Determining the Sequence Limit

Sequence limits have long existed in mathematics. Of course, they deserve their own competent design. So, time to learn the definition of sequence limits. First, let's look at the limit for a linear function in detail:

1. All limits are abbreviated as lim.
2. The notation of a limit consists of the abbreviation lim, any variable tending to a certain number, zero or infinity, as well as the function itself.

It is easy to understand that the definition of the limit of a sequence can be formulated as follows: this is a certain number to which all members of the sequence infinitely approach. A simple example: a x = 4x+1. Then the sequence itself will look like this.

5, 9, 13, 17, 21…x…

Thus, this sequence will increase indefinitely, which means its limit is equal to infinity as x→∞, and it should be written like this:

If we take a similar sequence, but x tends to 1, we get:

And the series of numbers will be like this: 1.4, 1.8, 4.6, 4.944, etc. Each time you need to substitute the number closer to one (0.1, 0.2, 0.9, 0.986). From this series it is clear that the limit of the function is five.

From this part it is worth remembering what the limit of a numerical sequence is, the definition and method for solving simple problems.

General designation for the limit of sequences

Having examined the limit of a number sequence, its definition and examples, you can proceed to a more complex topic. Absolutely all limits of sequences can be formulated by one formula, which is usually analyzed in the first semester.

So, what does this set of letters, modules and inequality signs mean?

∀ is a universal quantifier, replacing the phrases “for all”, “for everything”, etc.

∃ is an existential quantifier, in this case it means that there is some value N belonging to the set of natural numbers.

A long vertical stick following N means that the given set N is “such that.” In practice, it can mean “such that”, “such that”, etc.

To reinforce the material, read the formula out loud.

Uncertainty and certainty of the limit

The method of finding the limit of sequences, which was discussed above, although simple to use, is not so rational in practice. Try to find the limit for this function:

If we substitute different values ​​of “x” (increasing each time: 10, 100, 1000, etc.), then we get ∞ in the numerator, but also ∞ in the denominator. This results in a rather strange fraction:

But is this really so? Calculating the limit of a number sequence in this case seems quite easy. It would be possible to leave everything as it is, because the answer is ready, and it was received under reasonable conditions, but there is another way specifically for such cases.

First, let's find the highest degree in the numerator of the fraction - this is 1, since x can be represented as x 1.

Now let's find the highest degree in the denominator. Also 1.

Let's divide both the numerator and the denominator by the variable to the highest degree. In this case, divide the fraction by x 1.

Next, we will find what value each term containing a variable tends to. In this case, fractions are considered. As x→∞, the value of each fraction tends to zero. When submitting your work in writing, you should make the following footnotes:

This results in the following expression:

Of course, the fractions containing x did not become zeros! But their value is so small that it is completely permissible not to take it into account in calculations. In fact, x will never be equal to 0 in this case, because you cannot divide by zero.

What is a neighborhood?

Suppose the professor has at his disposal a complex sequence, given, obviously, by an equally complex formula. The professor has found the answer, but is it right? After all, all people make mistakes.

Auguste Cauchy once came up with an excellent way to prove the limits of sequences. His method was called neighborhood manipulation.

Suppose that there is a certain point a, its neighborhood in both directions on the number line is equal to ε (“epsilon”). Since the last variable is distance, its value is always positive.

Now let's define some sequence x n and assume that the tenth term of the sequence (x 10) is included in the neighborhood of a. How can we write this fact in mathematical language?

Let's say x 10 is to the right of point a, then the distance x 10 -a<ε, однако, если расположить «икс десятое» левее точки а, то расстояние получится отрицательным, а это невозможно, значит, следует занести левую часть неравенства под модуль. Получится |х 10 -а|<ε.

Now it’s time to explain in practice the formula discussed above. It is fair to call a certain number a the end point of a sequence if for any of its limits the inequality ε>0 is satisfied, and the entire neighborhood has its own natural number N, such that all members of the sequence with higher numbers will be inside the sequence |x n - a|< ε.

With such knowledge it is easy to solve the sequence limits, prove or disprove the ready-made answer.

Theorems

Theorems on the limits of sequences are an important component of the theory, without which practice is impossible. There are only four main theorems, remembering which can make the solution or proof much easier:

1. Uniqueness of the limit of a sequence. Any sequence can have only one limit or none at all. The same example with a queue that can only have one end.
2. If a series of numbers has a limit, then the sequence of these numbers is limited.
3. The limit of the sum (difference, product) of sequences is equal to the sum (difference, product) of their limits.
4. The limit of the quotient of dividing two sequences is equal to the quotient of the limits if and only if the denominator does not vanish.

Proof of sequences

Sometimes you need to solve an inverse problem, to prove a given limit of a numerical sequence. Let's look at an example.

Prove that the limit of the sequence given by the formula is zero.

According to the rule discussed above, for any sequence the inequality |x n - a|<ε. Подставим заданное значение и точку отсчёта. Получим:

Let us express n through “epsilon” to show the existence of a certain number and prove the presence of a limit of the sequence.

At this point, it is important to remember that “epsilon” and “en” are positive numbers and are not equal to zero. Now it is possible to continue further transformations using the knowledge about inequalities gained in high school.

How does it turn out that n > -3 + 1/ε. Since it is worth remembering that we are talking about natural numbers, the result can be rounded by putting it in square brackets. Thus, it was proven that for any value of the “epsilon” neighborhood of the point a = 0, a value was found such that the initial inequality is satisfied. From here we can safely say that the number a is the limit of a given sequence. Q.E.D.

This convenient method can be used to prove the limit of a numerical sequence, no matter how complex it may be at first glance. The main thing is not to panic when you see the task.

Or maybe he's not there?

The existence of a consistency limit is not necessary in practice. You can easily come across series of numbers that really have no end. For example, the same “flashing light” x n = (-1) n. it is obvious that a sequence consisting of only two digits, repeated cyclically, cannot have a limit.

The same story is repeated with sequences consisting of one number, fractional ones, having uncertainty of any order during calculations (0/0, ∞/∞, ∞/0, etc.). However, it should be remembered that incorrect calculations also occur. Sometimes double-checking your own solution will help you find the sequence limit.

Monotonic sequence

Several examples of sequences and methods for solving them were discussed above, and now let’s try to take a more specific case and call it a “monotonic sequence.”

Definition: any sequence can rightly be called monotonically increasing if the strict inequality x n holds for it< x n +1. Также любую последовательность справедливо называть монотонной убывающей, если для неё выполняется неравенство x n >x n +1.

Along with these two conditions, there are also similar non-strict inequalities. Accordingly, x n ≤ x n +1 (non-decreasing sequence) and x n ≥ x n +1 (non-increasing sequence).

But it’s easier to understand this with examples.

The sequence given by the formula x n = 2+n forms the following series of numbers: 4, 5, 6, etc. This is a monotonically increasing sequence.

And if we take x n =1/n, we get the series: 1/3, ¼, 1/5, etc. This is a monotonically decreasing sequence.

Limit of a convergent and bounded sequence

A bounded sequence is a sequence that has a limit. A convergent sequence is a series of numbers that has an infinitesimal limit.

Thus, the limit of a bounded sequence is any real or complex number. Remember that there can only be one limit.

The limit of a convergent sequence is an infinitesimal (real or complex) quantity. If you draw a sequence diagram, then at a certain point it will seem to converge, tend to turn into a certain value. Hence the name - convergent sequence.

Limit of a monotonic sequence

There may or may not be a limit to such a sequence. First, it is useful to understand when it exists; from here you can start when proving the absence of a limit.

Among monotonic sequences, convergent and divergent are distinguished. Convergent is a sequence that is formed by the set x and has a real or complex limit in this set. Divergent is a sequence that has no limit in its set (neither real nor complex).

Moreover, the sequence converges if, in a geometric representation, its upper and lower limits converge.

The limit of a convergent sequence can be zero in many cases, since any infinitesimal sequence has a known limit (zero).

Whatever convergent sequence you take, they are all bounded, but not all bounded sequences converge.

The sum, difference, product of two convergent sequences is also a convergent sequence. However, the quotient can also be convergent if it is defined!

Various actions with limits

Sequence limits are as significant (in most cases) as digits and numbers: 1, 2, 15, 24, 362, etc. It turns out that some operations can be performed with limits.

First, like digits and numbers, the limits of any sequence can be added and subtracted. Based on the third theorem on the limits of sequences, the following equality holds: the limit of the sum of sequences is equal to the sum of their limits.

Secondly, based on the fourth theorem on the limits of sequences, the following equality is true: the limit of the product of the nth number of sequences is equal to the product of their limits. The same is true for division: the limit of the quotient of two sequences is equal to the quotient of their limits, provided that the limit is not zero. After all, if the limit of sequences is equal to zero, then division by zero will result, which is impossible.

Properties of sequence quantities

It would seem that the limit of the numerical sequence has already been discussed in some detail, but phrases such as “infinitely small” and “infinitely large” numbers are mentioned more than once. Obviously, if there is a sequence 1/x, where x→∞, then such a fraction is infinitesimal, and if the same sequence, but the limit tends to zero (x→0), then the fraction becomes an infinitely large value. And such quantities have their own characteristics. The properties of the limit of a sequence having any small or large values ​​are as follows:

1. The sum of any number of any number of small quantities will also be a small quantity.
2. The sum of any number of large quantities will be an infinitely large quantity.
3. The product of arbitrarily small quantities is infinitesimal.
4. The product of any number of large numbers is infinitely large.
5. If the original sequence tends to an infinitely large number, then its inverse will be infinitesimal and tend to zero.

In fact, calculating the limit of a sequence is not such a difficult task if you know a simple algorithm. But the limits of consistency are a topic that requires maximum attention and perseverance. Of course, it is enough to simply grasp the essence of the solution to such expressions. Starting small, you can achieve great heights over time.