Formulate a definition in terms of cosha lim. Universal definition of the limit of a function by gain and by coch. Finite function limits at endpoints

Consider a 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 limit according to Cauchy

The number %%A \in \mathbb(R)%% is called function limit%%f(x)%% at %%a \in \mathbb(R)%% (or as %%x%% tends to %%a \in \mathbb(R)%%) if, what whatever the positive number %%\varepsilon%% is, there is a positive number %%\delta%% such that for all points of the punctured %%\delta%% neighborhood of the point %%a%% the values ​​of the function belong to %%\varepsilon %%-neighbourhood of the 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%% language definition, proposed by the French mathematician Augustin Cauchy and used with early XIX century to the present, because it has the necessary mathematical rigor and accuracy.

Combining different neighborhoods of the point %%a%% like %%\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 neighborhoods %%\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 sense

The geometric meaning of the limit of a function

Let us find out what is the geometric meaning of the limit of a function at a point. Let's plot 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%%-neighbourhood of the point %%A%% one can specify such a %%\ delta%%-neighborhood of the point %%a%% such that for any %%x%% of this %%\delta%%-neighborhood the value %%f(x)%% will be in the %%\varepsilon%%-neighborhood points %%A%%.

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

Definition 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, the sequence of corresponding values ​​%%\big\(f(x_n)\big\)%% tends to %%A%%.

The definition of the limit according to Heine is convenient to use when there are doubts about the existence of the 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 the limit of the function %%f(x)%% also does not exist.

Example

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

We 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 the sequence $$ 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 condition of the Heine definition, i.e. this function has no limit at the point %%x = 0%%.

Theorem

The definition of the limit according to Cauchy and according to Heine are equivalent.

Let's start with general things that are VERY important, but few people pay attention to them.

Function limit - basic concepts.

Infinity denote symbol . In fact, infinity is either an infinitely large positive number, or an infinitely large a negative number.

What it means: when you see , it doesn't matter if it is or . But it's better not to replace with , just as it's better not to replace with .

Write the limit of a function f(x) taken in the form, the argument x is indicated below and through the arrow to what value it tends.

If it is a specific real number, then they say about limit of a function at a point.

If or . then they talk about function limit at infinity.

The limit itself may be equal to a particular real number, in which case it is said that limit is finite.

If a , or , then they say that the limit is infinite.

They also say that limit does not exist, if it is impossible to determine the specific value of the limit or its infinite value (, or ). For example, there is no limit from sine to infinity.

Limit of a function - basic definitions.

It's time to get busy finding the values ​​of the limits of functions at infinity and at a point. A few definitions will help us with this. These definitions are based on number sequences and their convergence or divergence.

Definition(finding the limit of a function at infinity).

The number A is called the limit of the function f (x) when, if for any infinitely large sequence of arguments of the function (infinitely large positive or negative), the sequence of values ​​​​of this function converges to A. Designated .

Comment.

The limit of the function f(x) at is infinite if for any infinitely large sequence of function arguments (infinitely large positive or negative), the sequence of values ​​of this function is infinitely large positive or infinitely large negative. Designated .

Example.

Using the definition of the limit for prove the equality .

Solution.

Let's write down the sequence of function values ​​for an infinitely large positive sequence of argument values.

Obviously, the terms of this sequence decrease monotonically to zero.

Graphic illustration.

Now let's write down the sequence of function values ​​for an infinitely large negative sequence of argument values.

The terms of this sequence also decrease monotonically towards zero, which proves the original equality.

Graphic illustration.

Example.

Find the limit

Solution.

Let's write down the sequence of function values ​​for an infinitely large positive sequence of argument values. For example, let's take .

The sequence of function values ​​in this case will be (blue dots on the graph)

Obviously, this sequence is an infinitely large positive one, therefore,

Now let's write down the sequence of function values ​​for an infinitely large negative sequence of argument values. For example, let's take .

The sequence of function values ​​in this case will be (green dots on the graph)

Obviously, this sequence converges to zero, therefore,

Graphic illustration

Answer:

Now let's talk about the existence and finding the limit of a function at a point. Everything is based on setting unilateral limits. One cannot do without calculating one-sided limits for .

Definition(finding the limit of the function on the left).

The number B is called the limit of the function f (x) on the left when, if for any sequence of arguments of the function converging to a, the values ​​\u200b\u200bof which remain less than a (), the sequence of values ​​​​of this function converges to B.

Denoted .

Definition(finding the limit of the function on the right).

The number B is called the limit of the function f (x) on the right when, if for any sequence of arguments of the function converging to a, the values ​​\u200b\u200bof which remain greater than a (), the sequence of values ​​​​of this function converges to B.

Denoted .

Definition(existence of a limit of a function at a point).

The limit of the function f(x) at a point a exists if there are limits on the left and right of a and they are equal to each other.

Comment.

The limit of the function f(x) at a point a is infinite if the limits on the left and right of a are infinite.

Let us explain these definitions with an example.

Example.

Prove the existence of a finite limit of a function at point . Find its value.

Solution.

We will start from the definition of the existence of the limit of a function at a point.

First, we show the existence of a limit on the left. To do this, take a sequence of arguments converging to , and . An example of such a sequence would be

In the figure, the corresponding values ​​are shown with green dots.

It is easy to see that this sequence converges to -2, so .

Second, we show the existence of a limit on the right. To do this, take a sequence of arguments converging to , and . An example of such a sequence would be

The corresponding sequence of function values ​​will look like

In the figure, the corresponding values ​​are shown as blue dots.

It is easy to see that this sequence also converges to -2, so .

By this we have shown that the limits on the left and on the right are equal, therefore, by definition, there is a limit of the function at the point , and

Graphic illustration.

We recommend continuing the study of the basic definitions of the theory of limits with the topic.

The definitions of the limit of a function according to Heine (in terms of sequences) and in terms of Cauchy (in terms of epsilon and delta neighborhoods) are given. The definitions are given in a universal form applicable to both bilateral and one-sided limits at finite and at infinity points. The definition that a point a is not a limit of a function is considered. Proof of the equivalence of the definitions according to Heine and according to Cauchy.

Content

See also: Neighborhood of a point
Determining the limit of a function at the end point
Determining the limit of a function at infinity

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

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

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

.

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

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

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

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

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

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

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

Unilateral and bilateral limits

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

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

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

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

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

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

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

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

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

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

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

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

Proof

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

Heine proof ⇒ Cauchy

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

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

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

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

The first part is proven.

Cauchy proof ⇒ Heine

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

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

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

The theorem has been proven.

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

See also:

In this article, we will explain what the limit of a function is. First, let us explain the general points that are very important for understanding the essence of this phenomenon.

The concept of a limit

In mathematics, the concept of infinity, denoted by the symbol ∞, is fundamentally important. It should be understood as an infinitely large + ∞ or an infinitely small - ∞ number. When we talk about infinity, we often mean both of these meanings at once, but the notation of the form + ∞ or - ∞ should not be replaced simply with ∞.

The function limit is written as lim x → x 0 f (x) . At the bottom, we write the main argument x, and use the arrow to indicate which value x 0 it will tend to. If the value x 0 is a specific real number, then we are dealing with the limit of the function at a point. If the value x 0 tends to infinity (it does not matter, ∞, + ∞ or - ∞), then we should talk about the limit of the function at infinity.

The limit is finite and infinite. If it is equal to a specific real number, i.e. lim x → x 0 f (x) = A , then it is called the finite limit, but if lim x → x 0 f (x) = ∞ , lim x → x 0 f (x) = + ∞ or lim x → x 0 f (x) = - ∞ , then infinite.

If we cannot define either a finite or an infinite value, this means that such a limit does not exist. An example of this case would be the limit of sine at infinity.

In this paragraph, we will explain how to find the value of the limit of a function at a point and at infinity. To do this, we need to introduce basic definitions and remember what numerical sequences are, as well as their convergence and divergence.

Definition 1

The number A is the limit of the function f (x) as x → ∞, if the sequence of its values ​​will converge to A for any infinitely large sequence of arguments (negative or positive).

The function limit is written as follows: lim x → ∞ f (x) = A .

Definition 2

As x → ∞, the limit of the function f(x) is infinite if the sequence of values ​​for any infinitely large sequence of arguments is also infinitely large (positive or negative).

The notation looks like lim x → ∞ f (x) = ∞ .

Example 1

Prove the equality lim x → ∞ 1 x 2 = 0 using the basic definition of a limit for x → ∞ .

Solution

Let's start by writing a sequence of values ​​of the function 1 x 2 for an infinitely large positive sequence of values ​​of the argument x = 1 , 2 , 3 , . . . , n , . . . .

1 1 > 1 4 > 1 9 > 1 16 > . . . > 1 n 2 > . . .

We see that the values ​​will gradually decrease, tending to 0 . See picture:

x = - 1 , - 2 , - 3 , . . . , - n , . . .

1 1 > 1 4 > 1 9 > 1 16 > . . . > 1 - n 2 > . . .

Here, too, one can see a monotonic decrease to zero, which confirms the correctness of the given in the equality condition:

Answer: The correctness of the given in the condition of equality is confirmed.

Example 2

Calculate the limit lim x → ∞ e 1 10 x .

Solution

Let's start, as before, by writing sequences of values ​​f (x) = e 1 10 x for an infinitely large positive sequence of arguments. For example, x = 1 , 4 , 9 , 16 , 25 , . . . , 10 2 , . . . → +∞ .

e 1 10 ; e 4 10 ; e 9 10 ; e 16 10 ; e 25 10 ; . . . ; e 100 10 ; . . . == 1 , 10 ; 1, 49; 2, 45; 4, 95; 12, 18; . . . ; 22026, 46; . . .

We see that this sequence is infinitely positive, so f (x) = lim x → + ∞ e 1 10 x = + ∞

We proceed to write the values ​​of an infinitely large negative sequence, for example, x = - 1 , - 4 , - 9 , - 16 , - 25 , . . . , - 10 2 , . . . → -∞ .

e - 1 10 ; e - 4 10 ; e - 9 10 ; e - 16 10 ; e - 25 10 ; . . . ; e - 100 10 ; . . . == 0 , 90 ; 0.67; 0, 40; 0, 20; 0, 08; . . . ; 0,000045; . . . x = 1 , 4 , 9 , 16 , 25 , . . . , 10 2 , . . . →∞

Since it also tends to zero, then f (x) = lim x → ∞ 1 e 10 x = 0 .

The solution of the problem is clearly shown in the illustration. The blue dots mark the sequence of positive values, the green dots mark the sequence of negative ones.

Answer: lim x → ∞ e 1 10 x = + ∞ , pr and x → + ∞ 0 , pr and x → - ∞ .

Let us pass to the method of calculating the limit of a function at a point. To do this, we need to know how to properly define the one-sided limit. This will also be useful to us in order to find the vertical asymptotes of the function graph.

Definition 3

The number B is the limit of the function f (x) on the left as x → a in the case when the sequence of its values ​​converges to a given number for any sequence of arguments of the function x n , converging to a , if its values ​​remain less than a (x n< a).

Such a limit is written in writing as lim x → a - 0 f (x) = B .

Now we formulate what is the limit of the function on the right.

Definition 4

The number B is the limit of the function f (x) on the right as x → a in the case when the sequence of its values ​​converges to a given number for any sequence of arguments of the function x n , converging to a , if its values ​​remain greater than a (x n > a) .

We write this limit as lim x → a + 0 f (x) = B .

We can find the limit of the function f (x) at some point when it has equal limits on the left and right sides, i.e. lim x → a f (x) = lim x → a - 0 f (x) = lim x → a + 0 f (x) = B . In the case of infinity of both limits, the limit of the function at the starting point will also be infinite.

Now we will explain these definitions by writing down the solution of a specific problem.

Example 3

Prove that there is a finite limit of the function f (x) = 1 6 (x - 8) 2 - 8 at the point x 0 = 2 and calculate its value.

Solution

In order to solve the problem, we need to recall the definition of the limit of a function at a point. First, let's prove that the original function has a limit on the left. Let's write down the sequence of function values ​​that will converge to x 0 = 2 if x n< 2:

f(-2) ; f(0) ; f (1) ; f 1 1 2 ; f 1 3 4 ; f 1 7 8 ; f 1 15 16 ; . . . ; f 1 1023 1024 ; . . . == 8 , 667 ; 2,667; 0, 167; - 0,958; - 1, 489; - 1, 747; - 1, 874; . . . ; - 1, 998; . . . → - 2

Since the above sequence reduces to - 2 , we can write that lim x → 2 - 0 1 6 x - 8 2 - 8 = - 2 .

6 , 4 , 3 , 2 1 2 , 2 1 4 , 2 1 8 , 2 1 16 , . . . , 2 1 1024 , . . . → 2

The function values ​​in this sequence will look like this:

f(6) ; f (4) ; f (3) ; f 2 1 2 ; f 2 3 4 ; f 2 7 8 ; f 2 15 16 ; . . . ; f 2 1023 1024 ; . . . == - 7, 333; - 5, 333; - 3, 833; - 2, 958; - 2, 489; - 2, 247; - 2, 124; . . . , - 2 , 001 , . . . → - 2

This sequence also converges to - 2 , so lim x → 2 + 0 1 6 (x - 8) 2 - 8 = - 2 .

We have obtained that the limits on the right and left sides of this function will be equal, which means that the limit of the function f (x) = 1 6 (x - 8) 2 - 8 exists at the point x 0 = 2, and lim x → 2 1 6 (x - 8) 2 - 8 = - 2 .

You can see the progress of the solution in the illustration (green dots are a sequence of values ​​converging to x n< 2 , синие – к x n > 2).

Answer: The limits on the right and left sides of this function will be equal, which means that the limit of the function exists, and lim x → 2 1 6 (x - 8) 2 - 8 = - 2 .

To study the theory of limits in more depth, we advise you to read the article about the continuity of a function at a point and the main types of discontinuity points.

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In proving the properties of the limit of a function, we made sure that nothing really was required from the punctured neighborhoods in which our functions were defined and which arose in the course of proofs, except for the properties indicated in the introduction to the previous paragraph 2. This circumstance serves as a justification for singling out the following mathematical object.

a. Base; definition and main examples

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

In other words, the elements of the 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 the analysis.

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

The record will be used instead of It means that a; tends over the set E to a, remaining greater (less) than a.

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

The notation will be used instead

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

Note that all the listed bases have the feature 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 meet with other bases when studying functions that are not given on the real axis.

We also note that the term “base” used here is a short designation of what is called “filter basis” in mathematics, 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. Base function limit

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 the base B if for any neighborhood of the point A there is an element of the base whose image is contained in the neighborhood

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

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

Since we are now considering 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), we take a neighborhood that is symmetric (with respect to the point A) (e-neighborhood). 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 (carry out the proof in full!).

We have given a general definition of the limit of a function with respect to the base. Above were considered examples of the most common bases in the 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 particular base.

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

or, which is the same,

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

In order to be considered proven in the general case of a limit over an arbitrary base, all those theorems on limits that we proved in Section 2 for a special base , it is necessary to give the appropriate definitions: finally constant, finally bounded, and infinitely small for a given base of functions.

Definition 13. A function is called finally constant at base B if there exists a number and such an element of the base, at any point of which

Definition 14. A function is called bounded at base B or finally bounded at base B if there exists a number c and such an element of the base, at any point of which

Definition 15. A function is called infinitesimal with base B if

After these definitions and the basic observation that only base properties are needed to prove limit theorems, we can assume that all limit properties established in Section 2 are valid for limits over any base.

In particular, we can now talk about the limit of a function at or at or at

In addition, we have secured the possibility of applying the theory of limits even in the case when the functions are not defined on numerical sets; this will prove to be especially valuable in the future. For example, the length of a curve is a numerical function defined on some class of curves. If we know this function on broken lines, then by passing to the limit we determine it for more complex curves, for example, for a circle.

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

In order to finally get used to the concept of a limit over an arbitrary base, we will prove the further properties of the limit of a function in a general form.