What is called the limit of a function. Universal definition of the limit of a function according to Hein and Cauchy. Infinitesimal and infinitely large functions

By proving the properties of the limit of a function, we were convinced that nothing was really required from the punctured neighborhoods in which our functions were defined and which arose in the process of proof, except for the properties indicated in the introduction to the previous paragraph 2. This circumstance serves as a justification for identifying the following mathematical object.

A. Base; definition and basic examples

Definition 11. A collection B of subsets of a set X will be called a base in the set X if two conditions are met:

In other words, the elements of collection B are non-empty sets, and the intersection of any two of them contains some element from the same collection.

Let us indicate some of the most commonly used bases in analysis.

If then instead they write and say that x tends to a from the right or from the side large values(respectively, on the left or on the side of smaller values). When a short record is accepted instead

The entry will be used instead of She means that a; tends over the set E to a, remaining greater (smaller) than a.

then instead they write and say that x tends to plus infinity (respectively, to minus infinity).

The entry will be used instead

When instead of (if this does not lead to a misunderstanding) we will, as is customary in the theory of the limit of a sequence, write

Note that all of the listed bases have the peculiarity that the intersection of any two elements of the base is itself an element of this base, and not only contains some element of the base. We will encounter other bases when studying functions that are not specified on the number axis.

Note also that the term “base” used here is a short designation of what is called in mathematics “filter basis”, and the base limit introduced below is the most essential part for analysis of the concept of filter limit created by the modern French mathematician A. Cartan

b. Function limit by base

Definition 12. Let be a function on the set X; B is a base in X. A number is called the limit of a function with respect to base B if for any neighborhood of point A there is an element of the base whose image is contained in the neighborhood

If A is the limit of a function with respect to base B, then write

Let us repeat the definition of the limit by base in logical symbolism:

Since we are now looking at functions with numeric values, it is useful to keep in mind the following form of this basic definition:

In this formulation, instead of an arbitrary neighborhood V (A), a symmetric (with respect to point A) neighborhood (e-neighborhood) is taken. The equivalence of these definitions for real-valued functions follows from the fact that, as already mentioned, any neighborhood of a point contains some symmetric neighborhood of the same point (perform the proof in full!).

We gave general definition limit of the function over the base. Above we discussed examples of the most commonly used databases in analysis. In a specific problem where one or another of these bases appears, it is necessary to be able to decipher the general definition and write it down for a specific base.

Considering examples of bases, we, in particular, introduced the concept of a neighborhood of infinity. If we use this concept, then in accordance with the general definition of the limit it is reasonable to accept the following conventions:

or, what is the same,

Usually we mean a small value. This is, of course, not the case in the above definitions. In accordance with accepted conventions, for example, we can write

In order to be considered proven in general case limit on an arbitrary base, all those theorems about limits that we proved in paragraph 2 for a special base, it is necessary to give the corresponding definitions: finally constant, finally limited and infinitesimal for a given base of functions.

Definition 13. A function is said to be finally constant with base B if there exists a number and an element of the base such that at any point

At the moment, the main benefit of the observation made and the concept of a base introduced in connection with it is that they save us from checks and formal proofs of limit theorems for each specific type of limit passages or, in our current terminology, for each specific type bases

In order to finally become familiar with the concept of a limit over an arbitrary base, we will carry out proofs of further properties of the limit of a function in a general form.

(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 we use as a left-sided punctured neighborhood of the end point, 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. Well mathematical analysis. Volume 1. Moscow, 2003.

Limits give all mathematics students a lot of trouble. To solve a limit, sometimes you have to use a lot of tricks and choose from a variety of solution methods exactly the one that is suitable for a particular example.

In this article we will not help you understand the limits of your capabilities or comprehend the limits of control, but we will try to answer the question: how to understand the limits in higher mathematics? Understanding comes with experience, so at the same time we will give a few detailed examples solutions of limits with explanations.

The concept of limit in mathematics

The first question is: what is this limit and the limit of what? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since this is what students most often encounter. But first, the most general definition of a limit:

Let's say there is some variable value. If this value in the process of change unlimitedly approaches a certain number a , That a – the limit of this value.

For a function defined in a certain interval f(x)=y such a number is called a limit A , which the function tends to when X , tending to a certain point A . Dot A belongs to the interval on which the function is defined.

It sounds cumbersome, but it is written very simply:

Lim- from English limit- limit.

There is also a geometric explanation for determining the limit, but here we will not delve into the theory, since we are more interested in the practical rather than the theoretical side of the issue. When we say that X tends to some value, this means that the variable does not take on the value of a number, but approaches it infinitely close.

Let's give specific example. The task is to find the limit.

To solve this example, we substitute the value x=3 into a function. We get:

By the way, if you are interested, read separate article about this theme.

In examples X can tend to any value. It can be any number or infinity. Here's an example when X tends to infinity:

It’s intuitively clear what’s what larger number in the denominator, the smaller the value the function will take. So, with unlimited growth X meaning 1/x will decrease and approach zero.

As you can see, to solve the limit, you just need to substitute the value to strive for into the function X . However, this is the simplest case. Often finding the limit is not so obvious. Within the limits there are uncertainties of the type 0/0 or infinity/infinity . What to do in such cases? Resort to tricks!

Uncertainties within

Uncertainty of the form infinity/infinity

Let there be a limit:

If we try to substitute infinity into the function, we will get infinity in both the numerator and the denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: you need to notice how you can transform the function in such a way that the uncertainty goes away. In our case, we divide the numerator and denominator by X in the senior degree. What will happen?

From the example already discussed above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:

To resolve type uncertainties infinity/infinity divide the numerator and denominator by X to the highest degree.

By the way! For our readers there is now a 10% discount on

Another type of uncertainty: 0/0

As always, substituting values ​​into the function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that in our numerator quadratic equation. Let's find the roots and write:

Let's reduce and get:

So, if you are faced with type uncertainty 0/0 – factor the numerator and denominator.

To make it easier for you to solve examples, we present a table with the limits of some functions:

L'Hopital's rule within

Another powerful way to eliminate both types of uncertainty. What is the essence of the method?

If there is uncertainty in the limit, take the derivative of the numerator and denominator until the uncertainty disappears.

L'Hopital's rule looks like this:

Important point : the limit in which the derivatives of the numerator and denominator stand instead of the numerator and denominator must exist.

And now - a real example:

There is typical uncertainty 0/0 . Let's take the derivatives of the numerator and denominator:

Voila, uncertainty is resolved quickly and elegantly.

We hope that you will be able to usefully apply this information in practice and find the answer to the question “how to solve limits in higher mathematics.” If you need to calculate the limit of a sequence or the limit of a function at a point, and there is absolutely no time for this work, contact a professional student service for a quick and detailed solution.

Definition 1. Let E- an infinite number. If any neighborhood contains points of the set E, different from the point A, That A called ultimate point of the set E.

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

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

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

2) For function

.

This is obvious, because if
, then
.

3) Dirichlet function
has no limit at any point.

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

4)
.

Indeed, let us take an arbitrary sequence
, converging to

number 2. Then . Q.E.D.

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

,

inequality is true

.

They write:
.

Cauchy's definition can also be given using neighborhoods, if we note that , a:

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

It is useful to illustrate this definition with a drawing.

Example 5.
.

Indeed, let's take
randomly and find
, such that for everyone X, satisfying the inequality
inequality holds
. The last inequality is equivalent to the inequality
, so we see that it is enough to take
. The statement has been proven.

Fair

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

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

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

according to Heine.

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

Let's assume the opposite, i.e. What
according to Cauchy. Then there is
such that for anyone
there will be
,
And
. Consider the sequence
. For the specified
and any n exists

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

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

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

Similar to the Cauchy criterion for sequences, the Cauchy criterion for the existence of a limit of a function holds. Before formulating it, let us give

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

, such that
And
, the inequality holds
.

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

Proof.Necessity. Let
. We must prove that
satisfies at the point Cauchy condition.

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

The need has been proven.

Adequacy. Let the function
satisfies at the point Cauchy condition. We must prove that it has at the point final limit.

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

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

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

Consider the function %%f(x)%% defined at least in some punctured neighborhood %%\stackrel(\circ)(\text(U))(a)%% of the point %%a \in \overline( \mathbb(R))%% extended number line.

The concept of a Cauchy limit

The number %%A \in \mathbb(R)%% is called limit of the function%%f(x)%% at the point %%a \in \mathbb(R)%% (or at %%x%% tending to %%a \in \mathbb(R)%%), if, what Whatever the positive number %%\varepsilon%%, there is a positive number %%\delta%% such that for all points in the punctured %%\delta%% neighborhood of the point %%a%% the function values ​​belong to %%\varepsilon %%-neighborhood of point %%A%%, or

$$ A = \lim\limits_(x \to a)(f(x)) \Leftrightarrow \forall\varepsilon > 0 ~\exists \delta > 0 \big(x \in \stackrel(\circ)(\text (U))_\delta(a) \Rightarrow f(x) \in \text(U)_\varepsilon (A) \big) $$

This definition is called the %%\varepsilon%% and %%\delta%% definition, proposed by the French mathematician Augustin Cauchy and used with early XIX century to the present, since it has the necessary mathematical rigor and accuracy.

Combining various neighborhoods of the point %%a%% of the form %%\stackrel(\circ)(\text(U))_\delta(a), \text(U)_\delta (\infty), \text(U) _\delta (-\infty), \text(U)_\delta (+\infty), \text(U)_\delta^+ (a), \text(U)_\delta^- (a) %% with surroundings %%\text(U)_\varepsilon (A), \text(U)_\varepsilon (\infty), \text(U)_\varepsilon (+\infty), \text(U) _\varepsilon (-\infty)%%, we get 24 definitions of the Cauchy limit.

Geometric meaning

Geometric meaning of the limit of a function

Let's find out what it is geometric meaning limit of a function at a point. Let's build a graph of the function %%y = f(x)%% and mark the points %%x = a%% and %%y = A%% on it.

The limit of the function %%y = f(x)%% at the point %%x \to a%% exists and is equal to A if for any %%\varepsilon%% neighborhood of the point %%A%% one can specify such a %%\ delta%%-neighborhood of the point %%a%%, such that for any %%x%% from this %%\delta%%-neighborhood the value %%f(x)%% will be in the %%\varepsilon%%-neighborhood points %%A%%.

Note that by the definition of the limit of a function according to Cauchy, for the existence of a limit at %%x \to a%%, it does not matter what value the function takes at the point %%a%%. Examples can be given where the function is not defined when %%x = a%% or takes a value other than %%A%%. However, the limit may be %%A%%.

Determination of the Heine limit

The element %%A \in \overline(\mathbb(R))%% is called the limit of the function %%f(x)%% at %% x \to a, a \in \overline(\mathbb(R))%% , if for any sequence %%\(x_n\) \to a%% from the domain of definition, the sequence of corresponding values ​​%%\big\(f(x_n)\big\)%% tends to %%A%%.

The definition of a limit according to Heine is convenient to use when doubts arise about the existence of a limit of a function at a given point. If it is possible to construct at least one sequence %%\(x_n\)%% with a limit at the point %%a%% such that the sequence %%\big\(f(x_n)\big\)%% has no limit, then we can conclude that the function %%f(x)%% has no limit at this point. If for two various sequences %%\(x"_n\)%% and %%\(x""_n\)%% having same limit %%a%%, sequences %%\big\(f(x"_n)\big\)%% and %%\big\(f(x""_n)\big\)%% have various limits, then in this case there is also no limit of the function %%f(x)%%.

Example

Let %%f(x) = \sin(1/x)%%. Let's check whether the limit of this function exists at the point %%a = 0%%.

Let us first choose a sequence $$ \(x_n\) = \left\(\frac((-1)^n)(n\pi)\right\) converging to this point. $$

It is clear that %%x_n \ne 0~\forall~n \in \mathbb(N)%% and %%\lim (x_n) = 0%%. Then %%f(x_n) = \sin(\left((-1)^n n\pi\right)) \equiv 0%% and %%\lim\big\(f(x_n)\big\) = 0 %%.

Then take a sequence converging to the same point $$ x"_n = \left\( \frac(2)((4n + 1)\pi) \right\), $$

for which %%\lim(x"_n) = +0%%, %%f(x"_n) = \sin(\big((4n + 1)\pi/2\big)) \equiv 1%% and %%\lim\big\(f(x"_n)\big\) = 1%%. Similarly for the sequence $$ x""_n = \left\(-\frac(2)((4n + 1) \pi) \right\), $$

also converging to the point %%x = 0%%, %%\lim\big\(f(x""_n)\big\) = -1%%.

All three sequences gave different results, which contradicts the Heine definition condition, i.e. this function has no limit at the point %%x = 0%%.

Theorem

The Cauchy and Heine definitions of the limit are equivalent.