Theoretical minimum

The concept of a limit as applied to numerical sequences has already been introduced in the topic "".
It is recommended that you read the material contained there first.

Turning to the subject of this topic, we recall the concept of a function. The function is another example of mapping. We will consider the simplest case
real function of one real argument(what is the complexity of other cases - will be said later). The function within this topic is understood as
the law according to which each element of the set on which the function is defined is assigned one or more elements
set called the set of function values. If each element of the scope of a function is associated with one element
set of values, then the function is called single-valued, otherwise the function is called multi-valued. Here, for simplicity, we will only talk about
unambiguous functions.

I would immediately like to emphasize the fundamental difference between a function and a sequence: the sets connected by the mapping in these two cases are essentially different.
To avoid the need to use the terminology of general topology, we explain the difference with the help of imprecise reasoning. When discussing the limit
sequences, we talked about only one option: the unlimited growth of the number of the element of the sequence. As the number increases, the elements themselves
sequences behaved much more differently. They could "accumulate" in a small neighborhood of a certain number; they could grow indefinitely, and so on.
Roughly speaking, the assignment of a sequence is the assignment of a function on a discrete "domain". If we talk about the function, the definition of which is given
at the beginning of the topic, then the concept of the limit should be built more carefully. It makes sense to talk about the limit of the function when its argument tends to a certain value .
Such a formulation of the question did not make sense in relation to sequences. There is a need to make some clarifications. All of them are related to
how exactly the argument tends to the value in question.

Let's look at a few examples - for now in passing:


These functions will allow us to consider a variety of cases. We present here the graphs of these functions for greater clarity of presentation.

The function has a limit at any point in the domain of definition - this is intuitively clear. Whatever point of the domain of definition we take,
you can immediately tell what value the function tends to when the argument tends to the selected value, and the limit will be finite, unless the argument
does not go to infinity. The graph of the function has a break. This affects the properties of the function at the break point, but from the point of view of the limit
this point is not highlighted. The function is already more interesting: at the point it is not clear what value of the limit to assign to the function.
If we approach the point on the right, then the function tends to one value, if on the left, the function tends to another value. In previous
examples were not. The function, when tending to zero, even on the left, even on the right, behaves the same way, tending to infinity -
in contrast to the function, which tends to infinity as the argument tends to zero, but the sign of infinity depends on how
side we come to zero. Finally, the function behaves at zero completely incomprehensibly.

We formalize the concept of a limit using the epsilon-delta language. The main difference from the definition of the sequence limit will be the need
prescribe the desire of the function argument to some value. This requires the notion of a limit point of a set, which is auxiliary in this context.
A point is called a limit point of a set if in any neighborhood contains an infinite number of points,
belonging to and different from . A little later it will become clear why such a definition is required.

So, the number is called the limit of the function at the point , which is the limit point of the set , on which is defined
function if

Let's analyze this definition one by one. Here we single out the parts related to the desire of the argument to the value and the desire of the function
to the value. One should understand the general meaning of the written statement, which can be approximately interpreted as follows.
The function tends to when , if taking a number from a sufficiently small neighborhood of the point , we will
get the value of the function from a sufficiently small neighborhood of the number . And the smaller will be the neighborhood of the point from which the values ​​are taken
argument, the smaller will be the neighborhood of the point where the corresponding values ​​of the function will fall.

Let us return again to the formal definition of the limit and read it in the light of what has just been said. A positive number limits the neighborhood
point from which we will take the values ​​of the argument. Moreover, the values ​​of the argument, of course, are from the scope of the function and do not coincide with the function itself.
dot: we are writing aspiration, not a coincidence! So if we take the value of the argument from the specified -neighborhood of the point,
then the value of the function will fall into the -neighborhood of the point .
Finally, we bring the definition together. No matter how small we choose -neighborhood of the point , there will always be such -neighborhood of the point ,
that when choosing the values ​​of the argument from it, we will get to the neighborhood of the point . Of course, the size is a neighborhood of a point in this case
depends on what neighborhood of the point was given. If the neighborhood of the value of the function is large enough, then the corresponding spread of values
argument will be large. With a decrease in the vicinity of the function value, the corresponding spread in the values ​​of the argument will also decrease (see Fig. 2).

It remains to clarify some details. First, the requirement that the point be a limit eliminates the need to care that the point
from -neighborhood generally belongs to the domain of the function. Secondly, participation in determining the limit of the condition means
that an argument can approach a value from either the left or the right.

For the case when the function argument tends to infinity, the concept of a limit point should be defined separately. called the limit
set point if for any positive number the interval contains an uncountable set
points from the set.

Let's get back to the examples. The function is not of particular interest to us. Let's take a closer look at other features.

Examples.

Example 1 The graph of the function has a kink.
Function despite the singularity at a point, it has a limit at this point. The singularity at zero is the loss of smoothness.

Example 2 One-sided limits.
The function at a point has no limit. As already noted, for the existence of a limit, it is required that when
on the left and on the right, the function aspired to the same value. This is obviously not the case here. However, one can introduce the notion of a one-sided limit.
If the argument tends to a given value from the side of larger values, then one speaks of a right-hand limit; if from the side of smaller values ​​-
about the left-hand limit.
In case of function
- right-hand limit However, we can give an example when the infinite fluctuations of the sine do not interfere with the existence of the limit (moreover, two-sided).
An example would be the function . The chart is below; understandably build it to the end in the neighborhood
origin is not possible. The limit at is equal to zero.

Remarks .
1. There is an approach to determining the limit of a function that uses the limit of a sequence - the so-called. definition of Heine. There, a sequence of points is constructed that converges to the required value
argument - then the corresponding sequence of function values ​​converges to the limit of the function for this argument value. Equivalence of Heine's definition and language definition
"epsilon-delta" is proven.
2. The case of functions of two or more arguments is complicated by the fact that for the existence of a limit at a point, it is required that the value of the limit is the same for any way the argument tends to
to the required value. If there is only one argument, then you can strive for the required value from the left or right. In the case of more variables, the number of options increases dramatically. The Case of Functions
complex variable and requires a separate discussion.

texvc - neighborhood set in functional analysis and related disciplines is such a set, each point of which is removed from given set no more than Unable to parse expression (executable file texvc not found; See math/README for setup help.): \varepsilon .

Definitions

• Let Unable to parse expression (executable file texvc not found; See math/README for setup help.): (X,\varrho) is a metric space, Unable to parse expression (executable file texvc not found; See math/README for setup help.): x_0 \in X, And Unable to parse expression (executable file texvc not found; See math/README for setup help.): \varepsilon > 0. Unable to parse expression (executable file texvc not found; See math/README for setup help.): \varepsilon-neighborhood Unable to parse expression (executable file texvc is called a set

Unable to parse expression (executable file texvc not found; See math/README for setup help.): U_(\varepsilon)(x_0) = \( x\in X \mid \varrho(x,x_0)< \varepsilon \}.

• Let a subset be given Unable to parse expression (executable file texvc not found; See math/README for setup help.): A \subset X. Then Unable to parse expression (executable file texvc not found; See math/README for setup help.): \varepsilon-neighborhood of this set is called the set

Unable to parse expression (executable file texvc not found; See math/README for setup help.): U_(\varepsilon)(A) = \bigcup\limits_(x \in A) U_(\varepsilon)(x).

Remarks

• Unable to parse expression (executable file texvc not found; See math/README for setup help.): \varepsilon-neighborhood of a point Unable to parse expression (executable file texvc not found; See math/README for setup help.): x_0 thus called an open ball centered at Unable to parse expression (executable file texvc not found; See math/README for setup help.): x_0 and radius Unable to parse expression (executable file texvc not found; See math/README for setup help.): \varepsilon.
• It follows directly from the definition that

Unable to parse expression (executable file texvc not found; See math/README for setup help.): U_(\varepsilon)(A) = \( x\in X \mid \exists y\in A\; \varrho(x,y)< \varepsilon\}.

• Unable to parse expression (executable file texvc not found; See math/README for setup help.): \varepsilon-neighborhood is a neighborhood and, in particular, an open set.

Examples

Write a review on the article "Epsilon neighborhood"

An excerpt characterizing the Epsilon neighborhood

- Well, what - listen? The little girl pushed me impatiently.
We came close... And I felt a wonderfully soft touch of a sparkling wave... It was something incredibly gentle, surprisingly affectionate and soothing, and at the same time, penetrating into the very "depth" of my surprised and slightly wary soul... Quiet “music” ran along my foot, vibrating in millions of different shades, and, rising up, began to envelop me with something fabulously beautiful, something that defied any words ... I felt that I was flying, although there was no flight was not real. It was wonderful!.. Each cell dissolved and melted in the oncoming new wave, and the sparkling gold washed right through me, taking away everything bad and sad and leaving only pure, primordial light in my soul...
I did not even feel how I entered and plunged into this sparkling miracle almost with my head. It was just incredibly good and I never wanted to leave there ...
- All right, that's enough already! We have a job ahead of us! Stella's assertive voice broke into the radiant beauty. - Did you like it?
- Oh, how! I breathed. - I didn't want to go out!
- Exactly! So some “bath” until the next incarnation ... And then they don’t come back here anymore ...

What icons besides inequality signs and modulus do you know?

From the course of algebra, we know the following notation:

- the universal quantifier means - "for any", "for all", "for each", that is, the entry should be read "for any positive epsilon";

– existential quantifier, – there is a value belonging to the set of natural numbers.

- a long vertical stick is read 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 sign of the modulus means the distance, i.e. this entry tells us that the distance between values ​​is less than epsilon.

Determining the Limit of a Sequence

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 the sequence:

It is easy to see that the subsequence is infinitely close to the number -1, and the even-numbered terms are close to "one".

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 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 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, except perhaps for a finite number of them.” This is closer to the truth, but still not entirely accurate. So, for example, in a 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 terms? The scientific world struggled with this problem for a long time, until the situation was resolved by the famous maestro, who, in essence, formalized classical mathematical analysis in all its rigor. Cauchy proposed to operate with neighborhoods, which significantly advanced the theory.

Consider some point and its arbitrary neighborhood:

The value of "epsilon" is always positive, and, moreover, we are free to choose it ourselves. Suppose that in a given neighborhood there is a set of members (not necessarily all) of 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 “a” point, then the difference will be negative, and therefore the module sign must be added to it: .

Definition: a number is called the limit of a sequence if for any of its neighborhoods (previously chosen) 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 will FULLY go 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 infinitely small.

It should be noted that for a sequence it is no longer possible to say “an infinite tail will enter” - members with odd numbers are in fact equal to zero and “do not go anywhere” =) That is why the verb “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 this neighborhood - odd members will always "jump" to "minus one". For a similar reason, there is no limit at the point .

Prove that the limit of the sequence is zero. Indicate 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 an arbitrary -neighborhood of the point and check if there is a number - such that ALL terms 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 the "school" actions with inequalities, which I repeated in the lessons Linear inequalities and Domain of definition of a function. 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 is generally fractional, it needs to 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 the inequality and recall that initially we considered an arbitrary -neighborhood, i.e. "epsilon" can be equal to any positive number.

Conclusion : for any arbitrarily small -neighborhood of the point, a value was found such that the inequality holds for all larger numbers. 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 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 a large, but finite number of members.

The general definition of a neighborhood of a point on the real line is considered. Definitions of epsilon neighborhoods, left-handed, right-handed, and punctured neighborhoods of endpoints and at infinity. Neighborhood property. A theorem on the equivalence of using an epsilon neighborhood and an arbitrary neighborhood in the definition of the Cauchy limit of a function is proved.

Content

Determination of the neighborhood of a point

A neighborhood of a real point x 0 Any open interval containing this point is called:
.
Here ε 1 and ε 2 are arbitrary positive numbers.

Epsilon - neighborhood of point x 0 is called the set of points, the distance from which to the point x 0 less than ε:
.

The punctured neighborhood of the point x 0 is called the neighborhood of this point, from which the point x itself has been excluded 0 :
.

Neighborhood endpoints

At the very beginning, the definition of a neighborhood of a point was given. It is designated as . But you can explicitly specify that a neighborhood depends on two numbers using the appropriate arguments:
(1) .
That is, a neighborhood is a set of points belonging to an open interval.

Equating ε 1 to ε 2 , we get epsilon - neighborhood:
(2) .
Epsilon - a neighborhood - is a set of points belonging to an open interval with equidistant ends.
Of course, the letter epsilon can be replaced by any other and we can consider δ - neighborhood, σ - neighborhood, and so on.

In the theory of limits, one can use the definition of a neighborhood based both on the set (1) and on the set (2). Using any of these neighborhoods gives equivalent results (see ). But the definition (2) is simpler, therefore, it is epsilon that is often used - the neighborhood of a point determined from (2).

The concepts of left-handed, right-handed, and punctured neighborhoods of endpoints are also widely used. We present their definitions.

Left-hand neighborhood of a real point x 0 is the half-open interval located on the real axis to the left of x 0 , including the dot itself:
;
.

Right-hand neighborhood of a real point x 0 is the half-open interval located to the right of x 0 , including the dot itself:
;
.

Punctured Endpoint Neighborhoods

Punctured neighborhoods of the point x 0 are the same neighborhoods, from which the point itself is excluded. They are identified with a circle above the letter. We present their definitions.

Punctured neighborhood of point x 0 :
.

Punctured epsilon - neighborhood of point x 0 :
;
.

Punctured left-hand neighborhood:
;
.

Punctured right-hand neighborhood:
;
.

Neighborhoods of points at infinity

Along with the end points, the notion of a neighborhood of points at infinity is also introduced. They are all punctured because there is no real number at infinity (at infinity is defined as the limit of an infinitely large sequence).

.
;
;
.

It was possible to determine the neighborhoods of infinitely distant points and so:
.
But instead of M, we use , so that a neighborhood with a smaller ε is a subset of a neighborhood with a larger ε , just like for neighborhoods of endpoints.

neighborhood property

Next, we use the obvious property of the neighborhood of a point (finite or at infinity). It lies in the fact that neighborhoods of points with smaller values ​​of ε are subsets of neighborhoods with larger values ​​of ε . We present more rigorous formulations.

Let there be a finite or infinitely distant point. Let it go .
Then
;
;
;
;
;
;
;
.

The converse assertions are also true.

Equivalence of definitions of the limit of a function according to Cauchy

Now we will show that in the definition of the limit of a function according to Cauchy, one can use both an arbitrary neighborhood and a neighborhood with equidistant ends .

Theorem
The Cauchy definitions of the limit of a function that use arbitrary neighborhoods and neighborhoods with equidistant ends are equivalent.

Proof

Let's formulate first definition of the limit of a function.
A number a is the limit of a function at a point (finite or at infinity) if for any positive numbers there exist numbers depending on and , such that for all , belongs to the corresponding neighborhood of the point a :
.

Let's formulate second definition of the limit of a function.
The number a is the limit of the function at the point , if for any positive number there exists a number depending on , such that for all :
.

Proof 1 ⇒ 2

Let us prove that if the number a is the limit of the function by the 1st definition, then it is also the limit by the 2nd definition.

Let the first definition hold. This means that there are such functions and , so for any positive numbers the following holds:
at , where .

Since the numbers and are arbitrary, we equate them:
.
Then there are functions and , so that for any the following holds:
at , where .

Notice, that .
Let be the smallest positive number and . Then, as noted above,
.
If , then .

That is, we found such a function , so that for any the following is true:
at , where .
This means that the number a is the limit of the function and by the second definition.

Proof 2 ⇒ 1

Let us prove that if the number a is the limit of the function by the 2nd definition, then it is also the limit by the 1st definition.

Let the second definition hold. Take two positive numbers and . And let be the smallest of them. Then, according to the second definition, there is such a function , so that for any positive number and for all , it follows that
.

But according to . Therefore, from what follows,
.

Then for any positive numbers and , we have found two numbers , so for all :
.

This means that the number a is also the limit by the first definition.

The theorem has been proven.

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