Let the function y=ƒ(x) be defined in some neighborhood of the point x o, except, perhaps, for the point x o itself.

Let us formulate two equivalent definitions of the limit of a function at a point.

Definition 1 (in the "language of sequences", or according to Heine).

The number A is called the limit of the function y \u003d ƒ (x) in the furnace x 0 (or at x® x o), if for any sequence of admissible values ​​of the argument x n, n є N (x n ¹ x 0) converging to x o the sequence of corresponding values ​​of the function ƒ(х n), n є N, converges to the number A

In this case, write
or ƒ(x)->A at x→x o. The geometric meaning of the limit of a function: means that for all points x sufficiently close to the point x o, the corresponding values ​​of the function differ arbitrarily little from the number A.

Definition 2 (in the "language of ε", or after Cauchy).

The number A is called the limit of the function at the point x o (or at x → x o) if for any positive ε there is a positive number δ such that for all x¹ x o satisfying the inequality |x-x o |<δ, выполняется неравенство |ƒ(х)-А|<ε.

The geometric meaning of the function limit:

if for any ε-neighbourhood of the point A there is such a δ-neighborhood of the point x o such that for all x¹ ho from this δ-neighborhood the corresponding values ​​of the function ƒ(x) lie in the ε-neighbourhood of the point A. In other words, the points of the graph of the function y = ƒ(x) lie inside a strip of width 2ε bounded by straight lines y=A+ ε , y=A-ε (see Fig. 110). Obviously, the value of δ depends on the choice of ε, so we write δ=δ(ε).

<< Пример 16.1

Prove that

Solution: Take an arbitrary ε>0, find δ=δ(ε)>0 such that for all x satisfying the inequality |х-3|< δ, выполняется неравенство |(2х-1)-5|<ε, т. е. |х-3|<ε.

Taking δ=ε/2, we see that for all x satisfying the inequality |x-3|< δ, выполняется неравенство |(2х-1)-5|<ε. Следовательно, lim(2x-1)=5 при х –>3.

<< Пример 16.2

16.2. One-sided limits

In the definition of the limit of the function, it is considered that x tends to x 0 in any way: remaining less than x 0 (to the left of x 0), greater than x o (to the right of x o), or fluctuating around the point x 0 .

There are cases when the method of approaching the argument x to xo significantly affects the value of the limit of the function. Therefore, the concept of one-sided limits is introduced.

The number A 1 is called the limit of the function y \u003d ƒ (x) on the left at the point x o, if for any number ε> 0 there is a number δ \u003d δ (ε)> 0 such that for x є (x 0 -δ; x o), the inequality |ƒ(x)-A|<ε. Предел слева записывают так: limƒ(х)=А при х–>x 0 -0 or briefly: ƒ (x o- 0) \u003d A 1 (Dirichlet notation) (see Fig. 111).

The limit of the function on the right is defined similarly, we write it using symbols:

Briefly, the limit on the right is denoted by ƒ(x o +0)=A.

The limits of a function on the left and right are called one-sided limits. Obviously, if exists, then both one-sided limits exist, and A=A 1 =A 2 .

The converse statement is also true: if both limits ƒ(x 0 -0) and ƒ(x 0 +0) exist and they are equal, then there is a limit and A \u003d ƒ(x 0 -0).

If A 1 ¹ A 2, then this aisle does not exist.

16.3. Limit of the function at x ® ∞

Let the function y=ƒ(x) be defined in the interval (-∞;∞). The number A is called function limitƒ(x) at x→ ∞ , if for any positive number ε there is such a number М=М()>0 that for all х satisfying the inequality |х|>М the inequality |ƒ(х)-А|<ε. Коротко это определение можно записать так:

The geometric meaning of this definition is as follows: for "ε>0 $ M>0, that for x є(-∞; -M) or x є(M; +∞) the corresponding values ​​of the function ƒ(x) fall into the ε-neighborhood of the point A , i.e., the points of the graph lie in a strip of width 2ε, bounded by straight lines y \u003d A + ε and y \u003d A-ε (see Fig. 112).

16.4. Infinitely large function (b.b.f.)

The function y=ƒ(x) is called infinitely large for x→x 0 if for any number M>0 there is a number δ=δ(M)>0, which for all x satisfying the inequality 0<|х-хо|<δ, выполняется неравенство |ƒ(х)|>M.

For example, the function y=1/(x-2) is a b.b.f. at x->2.

If ƒ(x) tends to infinity as x→x o and takes only positive values, then we write

if only negative values, then

The function y \u003d ƒ (x), given on the entire number line, called infinite for x→∞, if for any number M>0 there is such a number N=N(M)>0 that for all x satisfying the inequality |x|>N, the inequality |ƒ(x)|>M is satisfied. Short:

For example, y=2x has a b.b.f. at x→∞.

Note that if the argument х, tending to infinity, takes only natural values, i.e., хєN, then the corresponding b.b.f. becomes an infinitely large sequence. For example, the sequence v n =n 2 +1, n є N, is an infinitely large sequence. Obviously, every b.b.f. in a neighborhood of the point x o is unbounded in this neighborhood. The converse is not true: an unbounded function may not be a b.b.f. (For example, y=xsinx.)

However, if limƒ(x)=A for x→x 0 , where A is a finite number, then the function ƒ(x) is bounded in the vicinity of the point x o.

Indeed, from the definition of the limit of the function it follows that for x → x 0 the condition |ƒ(x)-A|<ε. Следовательно, А-ε<ƒ(х)<А+ε при х є (х о -ε; х о +ε), а это и означает, что функция ƒ (х) ограничена.

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

3) Dirichlet function
has no limit at any point.

Indeed, let
And
, and all are rational numbers. Then
for all n, That's why
. If
and 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, 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

,

the inequality

.

Write:
.

The definition of Cauchy can also be given with the help of neighborhoods, if you notice that , a:

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

It is useful to illustrate this definition with a figure.

Example 5.
.

Indeed, let's take
arbitrarily and find
, such that for all X satisfying the inequality
the inequality
. The last inequality is equivalent to the inequality
, so we see that it suffices to take
. The assertion 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
by Cauchy. Let us prove that the same number is also the limit according to Heine.

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

according to Heine.

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

Assume the opposite, i.e. What
by Cauchy. Then there is
such that for any
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 assertion. 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 it is the only one.

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

Similarly to the Cauchy criterion for sequences, there is a Cauchy criterion for the existence of a limit of a function. Before formulating it, we give

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

, such that
And
, the inequality
.

Theorem 3 (Cauchy's 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 have to prove that
satisfies at the point the 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
And
. Then

The need has been proven.

Adequacy. Let the function
satisfies at the point the Cauchy condition. Gotta prove that she has a point end limit.

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

Let us first show that for any sequence
, converging to , subsequence
function values ​​converge. 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. Assume the opposite, i.e. what are sequences
And
,
,
, such that. Let's consider a sequence. It is clear that it converges to , therefore, by the above, the sequence converges, which is impossible, since the subsequences
And
have different limits And . The obtained contradiction shows that =. Therefore, by Heine's definition, a function has at a point end limit. The sufficiency, and hence the theorem, are proved.

(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 reduces 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 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 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 is a 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.

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 of large values ​​(respectively, from 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 we use this concept, then in accordance with the general definition of the 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 about 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

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.

Today we will consider a selection of new problems for finding the limit at a point. Let's start with simple examples of value substitution, most often considered in the 11th grade of the school curriculum in mathematics.
Next, we stop and analyze the limits with uncertainties, methods for disclosing uncertainties, the application of the first and second important limits and their consequences.
The examples given will not fully cover the entire topic, but many questions will be clarified.

Find the limit of a function at a point:

Example 46. The limit of a function at a point is determined by substitution

Since the denominator of the fraction does not turn into zero, each graduate of the school can solve such a problem.

Example 47
Another task, actually for the 11th grade.

Example 48. Using the substitution method, we determine the limit of the function
It follows from the condition that the boundary of the function is equal to two if the variable tends to infinity.

Example 49. Direct substitution x=2 shows that the boundary at the point has the singularity (0/0) . This means that both the numerator and denominator implicitly contain (x-2) .
We perform the expansion of polynomials into simple factors, and then we reduce the fraction by the specified factor (x-2).
The limit of the fraction that remains is found by the substitution method.

Example 50. The limit of a function at a point has a singularity of type (0/0) .
We get rid of the difference in the roots by multiplying by the sum of the roots (adjoint expression), we expand the polynomial.
Further, simplifying the function, we find the value of the limit in unity.

Example 51. Consider a problem with complex limits.
So far, irrationality has been eliminated by multiplying by the conjugate expression.
Here, in the denominator, we have a cube root, so you need to use the formula for the difference of cubes.
All other transformations are repeated from condition to condition.
We decompose the polynomial into prime factors,
then we reduce by a factor that introduces a singularity (0)
and substituting x=-3 we find the limit of the function at the point

Example 52. We reveal a singularity of the form (0/0) using the first remarkable limit and its consequences.
First, we write the difference of the sines according to the trigonometric formula
sin(7x)-sin(3x)=2sin(2x)cos(5x).
Further, we supplement the numerator and denominator of the fraction with expressions that are necessary to highlight important limits.
We pass to the product of limits and evaluate the embedding of each factor.

Here we used the first remarkable limit:

and consequences from it


where a and b are arbitrary numbers.

Example 53. To reveal the uncertainty when the variable tends to zero, we use the second wonderful limit.
To isolate the exponent, we bring the exponent to the 2nd wonderful limit, and everything else that remains in the limit transition will give the degree of the exponent.


Here we used the corollary from the second remarkable limit:

Calculate the limit of a function at a point:

Example 54. You need to find the limit of a function at a point. A simple value substitution shows that we have a division of zeros.
To reveal it, we decompose the polynomials into simple factors and perform the reduction by the factor that introduces the singularity (x + 2) .
However, the numerator further contains (x+2) , which means that at x=-2 the boundary is zero.

Example 55. We have a fractional function - in the numerator the difference of the roots, in the denominator - a log.
Direct substitution gives a singularity of the form (0/0) .
The variable tends to minus one, which means that you should look for and get rid of features of the form (x+1) .
To do this, we get rid of irrationality by multiplying by the sum of the roots, and decompose the quadratic function into simple factors.
After all the reductions, by the substitution method, we determine the limit of the function at the point

Example 56. From the appearance of the sublimit function, one might mistakenly conclude that the first limit should be applied, but the calculations showed that everything is much simpler.
First write the sum of the sines in the denominator sin(2x)+sin(6x)=2sin(4x)*cos(2x).
Next, we paint tg(2x) , and the sine of the double angle sin(4x)=2sin(2x)cos (2x).
We simplify the sines and calculate the limit of the fraction by the substitution method

Example 57. The task of the ability to use the second wonderful limit:
the bottom line is that you should select the part that gives the exponent.
The rest that remains in the exponent in the passage to the limit will give the degree of the exponent.


The analysis of tasks into the limits of functions and sequences does not end there.
More than 150 ready answers to the limits of functions, so study and share links to materials with classmates.