MA. Function limit

texvc -neighborhood sets in functional analysis and related disciplines are such a set, each point of which is distant 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) there is a metric space, Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): 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-surroundings Unable to parse expression (Executable file texvc 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-the neighborhood of this set is 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).

Notes

• Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \varepsilon-neighborhood of the point Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): x_0 thus an open ball with center at is called Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): 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

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An excerpt characterizing the Epsilon neighborhood

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The general definition of a neighborhood of a point on the number line is considered. Definitions of epsilon neighborhood, left-sided, right-sided and punctured neighborhoods of finite and infinite points. Neighborhood property. A theorem is proven about the equivalence of the use of an epsilon neighborhood and an arbitrary neighborhood in determining the limit of a function according to Cauchy.

Content

Determining the neighborhood of a point

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

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

A punctured neighborhood of point x 0 is the neighborhood of this point from which the point x itself is excluded 0 :
.

Neighborhoods of endpoints

At the very beginning, a definition of the neighborhood of a point was given. It is designated as . But you can explicitly indicate that the 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) .
An epsilon 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 consider δ - neighborhood, σ - neighborhood, etc.

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

The concepts of left-sided, right-sided and punctured neighborhoods of endpoints are also widely used. Here are their definitions.

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

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

Punctured neighborhoods of endpoints

Punctured neighborhoods of point x 0 - these are the same neighborhoods from which the point itself is excluded. They are indicated with a circle above the letter. Here are their definitions.

Punctured neighborhood of point x 0 :
.

Punctured epsilon - neighborhood of point x 0 :
;
.

Pierced left side vicinity:
;
.

Punctured right side vicinity:
;
.

Neighborhoods of points at infinity

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

.
;
;
.

It was possible to determine the neighborhoods of points at infinity like this:
.
But instead of M, we use , so that the neighborhood with smaller ε is a subset of the neighborhood with larger ε, as for endpoint neighborhoods.

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. Here are more strict formulations.

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

The converse is also true.

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

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

Theorem
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 are numbers depending on and that for all belongs to the corresponding neighborhood of the point a:
.

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

Proof 1 ⇒ 2

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

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

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

Notice, that .
Let be the smallest of the positive numbers and . Then, according to what was noted above,
.
If, then.

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

Proof 2 ⇒ 1

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

Let the second definition be satisfied. Let's take two positive numbers and . And let it be the least 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 that
.

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

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

The theorem has been proven.

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

Theoretical minimum

The concept of a limit in relation to number sequences has already been introduced in the topic "".
It is recommended that you first read the material contained therein.

Moving on to the subject of this topic, let us recall the concept of function. The function is another example of mapping. We will consider the simplest case
real function of one real argument (what is difficult in other cases will be discussed later). The function within this topic is understood as
a 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 function's domain of definition is assigned one element
set of values, then the function is called single-valued, otherwise the function is called multi-valued. 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 a mapping in these two cases are significantly different.
To avoid the need to use the terminology of general topology, we will clarify the difference using imprecise reasoning. When discussing the limit
sequences, we talked about only one option: unlimited growth of the sequence element number. With this increase in number, the elements themselves
the sequences behaved much more diversely. They could “accumulate” in a small neighborhood of a certain number; they could grow unlimitedly, etc.
Roughly speaking, specifying a sequence is specifying a function on a discrete “domain of definition.” If we talk about a function, the definition of which is given
at the beginning of the topic, the concept of limit should be constructed more carefully. It makes sense to talk about the limit of the function when her argument tends to a certain value .
This 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 strives for the meaning in question.

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


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

A function at any point in its domain of definition has a limit - 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 if only the argument
does not tend to infinity. The graph of the function has a kink. 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 a point from the right, then the function tends to one value, if from the left, the function tends to another value. In previous
there were no examples of this. When a function tends to zero, either from the left or from the right, it 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 with what
side we are approaching zero. Finally, the function behaves completely incomprehensibly at zero.

Let's formalize the concept of a limit using the "epsilon-delta" language. The main difference from the definition of a sequence limit will be the need
describe the tendency of a function argument to a certain value. This requires the concept 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 countless 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 it is defined
function if

Let's look at this definition one by one. Let us highlight here the parts associated with the desire of the argument for meaning and with the desire of the function
to value . You should understand the general meaning of the written statement, which can be approximately interpreted as follows.
The function tends to at , if taking a number from a sufficiently small neighborhood of the point , we will
obtain the value of a function from a sufficiently small neighborhood of the number. And the smaller the neighborhood of the point from which the values ​​are taken
argument, the smaller will be the neighborhood of the point in which the corresponding function values ​​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 domain of definition of the function and do not coincide with the function itself
full stop: 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 in the -neighborhood of the point .
Finally, let's put the definition together. No matter how small we choose the -neighborhood of the point, there will always be such a -neighborhood of the point,
that when choosing the values ​​of the argument from it we will find ourselves in the vicinity of the point . Of course, the size is the neighborhood of the point in this case
depends on what neighborhood of the point was specified. If the neighborhood of the function value is large enough, then the corresponding spread of values
the argument will be great. As the neighborhood of the function value decreases, the corresponding spread of the argument values ​​will also decrease (see Fig. 2).

It remains to clarify some details. First, the requirement that a point be a limit eliminates the need to worry about whether a point
from the -neighborhood generally belongs to the domain of definition of the function. Secondly, participation in determining the limit condition means
that an argument can tend to a value both on the left and on the right.

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

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

Examples.

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

Example 2. One-sided limits.
A function at a point has no limit. As already noted, for the existence of a limit it is required that, when tending
on the left and on the right the function tended to the same value. This obviously doesn't hold here. However, the concept of a one-sided limit can be introduced.
If the argument tends to a given value from the side of larger values, then we speak of a right-handed limit; if on the side of smaller values ​​-
about the left-hand limit.
In case of function
- right-handed limit However, we can give an example when endless oscillations of the sine do not interfere with the existence of a limit (and a two-sided one).
An example would be the function . The graph is given below; for obvious reasons, build it to completion in the vicinity
origin is impossible. The limit at is zero.

Notes.
1. There is an approach to determining the limit of a function that uses the limit of a sequence - the so-called. Heine's definition. 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 at this argument value. Equivalence of Heine's definition and the definition in language
"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 be the same for any way the argument tends
to the required value. If there is only one argument, then you can strive for the required value from the left or from the right. When more variables, the number of options increases sharply. The case of functions
complex variable requires a separate discussion.

What symbols besides inequality signs and modulus do you know?

From the algebra course we know the following notation:

– the universal quantifier means “for any”, “for all”, “for everyone”, 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 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.

Determining the Sequence Limit

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 light of a practical lesson: “the limit of a sequence is the number to which the members of the sequence approach infinitely close.”

Okay, let's write down the sequence:

It is not difficult to grasp that the subsequence approaches the number –1 infinitely close, and the terms with even numbers approach “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 the only one.

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, with the possible exception of their finite number.” This is closer to the truth, but still not entirely accurate. So, for example, half of the terms of a sequence 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 symbols? 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 suggested operating in the surrounding area, which significantly advanced the theory.

Consider a certain point and its arbitrary neighborhood:

The value of “epsilon” is always positive, and, moreover, we have the right to choose it ourselves. Let us assume that in a given neighborhood there are many members (not necessarily all) of 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 modulus sign must be added to it: .

Definition: a number is called the limit of a sequence if for any of its neighborhoods (pre-selected) there is a natural number SUCH that ALL members of the sequence with larger 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.

So, for example, the “infinite tail” of the sequence will COMPLETELY go into any arbitrarily small -neighborhood of the point. Thus, this value is the limit of the sequence by definition. Let me remind you that a sequence whose limit is zero is called infinitesimal.

It should be noted that for a sequence it is no longer possible to say “an endless tail will come” - terms with odd numbers are in fact equal to zero and “will not 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.

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 an arbitrary -neighborhood of a point and check whether 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 it through .

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

We use “school” actions with inequalities, which I repeated in the lessons Linear inequalities and Domain 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 in general case is fractional, then 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 the inequality and remember that initially we considered an arbitrary -neighborhood, i.e. "epsilon" can be equal to any positive number.

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

By the way, a natural pattern is clearly visible from the result obtained: 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 a large, but finite number of terms.