Write down the definition of limit. Limits in mathematics for dummies: explanation, theory, examples of solutions

(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 is a 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. Course of mathematical analysis. Volume 1. Moscow, 2003.

Constant number A called limit sequences(x n ), if for any arbitrarily small positive numberε > 0 there is a number N that has all the values x n, for which n>N, satisfy the inequality

|x n - a|< ε. (6.1)

Write it down as follows: or x n → a.

Inequality (6.1) is equivalent to the double inequality

a- ε< x n < a + ε, (6.2)

which means that the points x n, starting from some number n>N, lie inside the interval (a-ε, a+ ε ), i.e. fall into any smallε -neighborhood of a point A.

A sequence having a limit is called convergent, otherwise - divergent.

The concept of a function limit is a generalization of the concept of a sequence limit, since the limit of a sequence can be considered as the limit of a function x n = f(n) of an integer argument n.

Let the function f(x) be given and let a - limit point domain of definition of this function D(f), i.e. such a point, any neighborhood of which contains points of the set D(f) other than a. Dot a may or may not belong to the set D(f).

Definition 1.The constant number A is called limit functions f(x) at x→a, if for any sequence (x n ) of argument values ​​tending to A, the corresponding sequences (f(x n)) have the same limit A.

This definition is called by defining the limit of a function according to Heine, or " in sequence language”.

Definition 2. The constant number A is called limit functions f(x) at x→a, if, by specifying an arbitrary arbitrarily small positive number ε, one can find such δ>0 (depending on ε), which is for everyone x, lying inε-neighborhoods of the number A, i.e. For x, satisfying the inequality
0 < x-a< ε , the values ​​of the function f(x) will lie inε-neighborhood of the number A, i.e.|f(x)-A|< ε.

This definition is called by defining the limit of a function according to Cauchy, or “in the language ε - δ “.

Definitions 1 and 2 are equivalent. If the function f(x) as x →a has limit, equal to A, this is written in the form

. (6.3)

In the event that the sequence (f(x n)) increases (or decreases) without limit for any method of approximation x to your limit A, then we will say that the function f(x) has infinite limit, and write it in the form:

A variable (i.e. a sequence or function) whose limit is zero is called infinitely small.

A variable whose limit is equal to infinity is called infinitely large.

To find the limit in practice, the following theorems are used.

Theorem 1 . If every limit exists

(6.4)

(6.5)

(6.6)

Comment. Expressions like 0/0, ∞/∞, ∞-∞ , 0*∞ , - are uncertain, for example, the ratio of two infinitesimal or infinitely large quantities, and finding a limit of this type is called “uncovering uncertainties.”

Theorem 2. (6.7)

those. one can go to the limit based on the power with a constant exponent, in particular, ;

(6.8)

(6.9)

Theorem 3.

(6.10)

(6.11)

Where e » 2.7 - base of natural logarithm. Formulas (6.10) and (6.11) are called the first wonderful limit and the second remarkable limit.

The consequences of formula (6.11) are also used in practice:

(6.12)

(6.13)

(6.14)

in particular the limit,

If x → a and at the same time x > a, then write x→a + 0. If, in particular, a = 0, then instead of the symbol 0+0 write +0. Similarly if x→a and at the same time x a-0. Numbers and are called accordingly right limit And left limit functions f(x) at the point A. For there to be a limit of the function f(x) as x→a is necessary and sufficient so that . The function f(x) is called continuous at the point x 0 if limit

. (6.15)

Condition (6.15) can be rewritten as:

,

that is, passage to the limit under the sign of a function is possible if it is continuous at a given point.

If equality (6.15) is violated, then we say that at x = x o function f(x) It has gap Consider the function y = 1/x. The domain of definition of this function is the set R, except for x = 0. The point x = 0 is a limit point of the set D(f), since in any neighborhood of it, i.e. in any open interval containing the point 0, there are points from D(f), but it itself does not belong to this set. The value f(x o)= f(0) is not defined, so at the point x o = 0 the function has a discontinuity.

The function f(x) is called continuous on the right at the point x o if the limit

,

And continuous on the left at the point x o, if the limit

.

Continuity of a function at a point xo is equivalent to its continuity at this point both to the right and to the left.

In order for the function to be continuous at a point xo, for example, on the right, it is necessary, firstly, that there be a finite limit, and secondly, that this limit be equal to f(x o). Therefore, if at least one of these two conditions is not met, then the function will have a discontinuity.

1. If the limit exists and is not equal to f(x o), then they say that function f(x) at the point x o has rupture of the first kind, or leap.

2. If the limit is+∞ or -∞ or does not exist, then they say that in point xo the function has a discontinuity second kind.

For example, function y = cot x at x→ +0 has a limit equal to +∞, which means that at the point x=0 it has a discontinuity of the second kind. Function y = E(x) (integer part of x) at points with whole abscissas has discontinuities of the first kind, or jumps.

A function that is continuous at every point in the interval is called continuous V . A continuous function is represented by a solid curve.

Many problems associated with the continuous growth of some quantity lead to the second remarkable limit. Such tasks, for example, include: growth of deposits according to the law of compound interest, growth of the country's population, decay of radioactive substances, proliferation of bacteria, etc.

Let's consider example of Ya. I. Perelman, giving an interpretation of the number e in the compound interest problem. Number e there is a limit . In savings banks, interest money is added to the fixed capital annually. If the accession is made more often, then the capital grows faster, since a larger amount is involved in the formation of interest. Let's take a purely theoretical, very simplified example. Let 100 deniers be deposited in the bank. units based on 100% per annum. If interest money is added to the fixed capital only after a year, then by this period 100 den. units will turn into 200 monetary units. Now let's see what 100 denize will turn into. units, if interest money is added to fixed capital every six months. After six months, 100 den. units will grow to 100× 1.5 = 150, and after another six months - at 150× 1.5 = 225 (den. units). If the accession is done every 1/3 of the year, then after a year 100 den. units will turn into 100× (1 +1/3) 3 " 237 (den. units). We will increase the terms for adding interest money to 0.1 year, to 0.01 year, to 0.001 year, etc. Then out of 100 den. units after a year it will be:

100 × (1 +1/10) 10 » 259 (den. units),

100 × (1+1/100) 100 » 270 (den. units),

100 × (1+1/1000) 1000 » 271 (den. units).

With an unlimited reduction in the terms for adding interest, the accumulated capital does not grow indefinitely, but approaches a certain limit equal to approximately 271. The capital deposited at 100% per annum cannot increase by more than 2.71 times, even if the accrued interest were added to the capital every second because the limit

Example 3.1.Using the definition of the limit of a number sequence, prove that the sequence x n =(n-1)/n has a limit equal to 1.

Solution.We need to prove that, no matter whatε > 0, no matter what we take, for it there is a natural number N such that for all n N the inequality holds|x n -1|< ε.

Let's take any e > 0. Since ; x n -1 =(n+1)/n - 1= 1/n, then to find N it is enough to solve the inequality 1/n< e. Hence n>1/ e and, therefore, N can be taken as an integer part of 1/ e , N = E(1/ e ). We have thereby proven that the limit .

Example 3.2 . Find the limit of a sequence given by a common term .

Solution.Let's apply the limit of the sum theorem and find the limit of each term. When n→ ∞ the numerator and denominator of each term tend to infinity, and we cannot directly apply the quotient limit theorem. Therefore, first we transform x n, dividing the numerator and denominator of the first term by n 2, and the second on n. Then, applying the limit of the quotient and the limit of the sum theorem, we find:

.

Example 3.3. . Find .

Solution. .

Here we used the limit of degree theorem: the limit of a degree is equal to the degree of the limit of the base.

Example 3.4 . Find ( ).

Solution.It is impossible to apply the limit of difference theorem, since we have an uncertainty of the form ∞-∞ . Let's transform the general term formula:

.

Example 3.5 . The function f(x)=2 1/x is given. Prove that there is no limit.

Solution.Let's use definition 1 of the limit of a function through a sequence. Let us take a sequence ( x n ) converging to 0, i.e. Let us show that the value f(x n)= behaves differently for different sequences. Let x n = 1/n. Obviously, then the limit Let us now choose as x n a sequence with a common term x n = -1/n, also tending to zero. Therefore there is no limit.

Example 3.6 . Prove that there is no limit.

Solution.Let x 1 , x 2 ,..., x n ,... be a sequence for which
. How does the sequence (f(x n)) = (sin x n) behave for different x n → ∞

If x n = p n, then sin x n = sin p n = 0 for all n and the limit If
x n =2 p n+ p /2, then sin x n = sin(2 p n+ p /2) = sin p /2 = 1 for all n and therefore the limit. So it doesn't exist.

Widget for calculating limits on-line

In the upper window, instead of sin(x)/x, enter the function whose limit you want to find. In the lower window, enter the number to which x tends and click the Calcular button, get the desired limit. And if in the result window you click on Show steps in the upper right corner, you will get a detailed solution.

Rules for entering functions: sqrt(x) - square root, cbrt(x) - cube root, exp(x) - exponent, ln(x) - natural logarithm, sin(x) - sine, cos(x) - cosine, tan (x) - tangent, cot(x) - cotangent, arcsin(x) - arcsine, arccos(x) - arccosine, arctan(x) - arctangent. Signs: * multiplication, / division, ^ exponentiation, instead infinity Infinity. Example: the function is entered as sqrt(tan(x/2)).

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.

Function y = f (x) is a law (rule) according to which each element x of the set X is associated with one and only one element y of the set Y.

Element x ∈ X called function argument or independent variable.
Element y ∈ Y called function value or dependent variable.

The set X is called domain of the function.
Set of elements y ∈ Y, which have preimages in the set X, is called area or set of function values.

The actual function is called limited from above (from below), if there is a number M such that the inequality holds for all:
.
The number function is called limited, if there is a number M such that for all:
.

Top edge or exact upper bound A real function is called the smallest number that limits its range of values ​​from above. That is, this is a number s for which, for everyone and for any, there is an argument whose function value exceeds s′: .
The upper bound of a function can be denoted as follows:
.

Respectively bottom edge or exact lower limit A real function is called the largest number that limits its range of values ​​from below. That is, this is a number i for which, for everyone and for any, there is an argument whose function value is less than i′: .
The infimum of a function can be denoted as follows:
.

Determining the limit of a function

Determination of the limit of a function according to Cauchy

Finite limits of function at end points

Let the function be defined in some neighborhood of the end point, with the possible exception of the point itself. at a point if for any there is such a thing, depending on , that for all x for which , the inequality holds
.
The limit of a function is denoted as follows:
.
Or at .

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

One-sided limits.
Left limit at a point (left-sided limit):
.
Right limit at a point (right-hand limit):
.
The left and right limits are often denoted as follows:
; .

Finite limits of a function at points at infinity

Limits at points at infinity are determined in a similar way.
.
.
.
They are often referred to as:
; ; .

Using the concept of neighborhood of a point

If we introduce the concept of a punctured neighborhood of a point, then we can give a unified definition of the finite limit of a function at finite and infinitely distant points:
.
Here for endpoints
; ;
.
Any neighborhood of points at infinity is punctured:
; ; .

Infinite Function Limits

Definition
Let the function be defined in some punctured neighborhood of a point (finite or at infinity). Limit of function f (x) as x → x 0 equals infinity, if for any arbitrarily large number M > 0 , there is a number δ M > 0 , depending on M, that for all x belonging to the punctured δ M - neighborhood of the point: , the following inequality holds:
.
The infinite limit is denoted as follows:
.
Or at .

Using the logical symbols of existence and universality, the definition of the infinite limit of a function can be written as follows:
.

You can also introduce definitions of infinite limits of certain signs equal to and :
.
.

Universal definition of the limit of a function

Using the concept of a neighborhood of a point, we can give a universal definition of the finite and infinite limit of a function, applicable both for finite (two-sided and one-sided) and infinitely distant points:
.

Determination of the limit of a function according to Heine

Let the function be defined on some set X:.
The number a is called the limit of the function at point:
,
if for any sequence converging to x 0 :
,
whose elements belong to the set X: ,
.

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

If we take the left-sided neighborhood of the point x as a set X 0 , then we obtain the definition of the left limit. If it is right-handed, then we get the definition of the right limit. If we take the neighborhood of a point at infinity as a set X, we obtain the definition of the limit of a function at infinity.

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

Properties and theorems of the limit of a function

Further, we assume that the functions under consideration are defined in the corresponding neighborhood of the point, which is a finite number or one of the symbols: . It can also be a one-sided limit point, that is, have the form or . The neighborhood is two-sided for a two-sided limit and one-sided for a one-sided limit.

Basic properties

If the values ​​of the function f (x) change (or make undefined) a finite number of points x 1, x 2, x 3, ... x n, then this change will not affect the existence and value of the limit of the function at an arbitrary point x 0 .

If there is a finite limit, then there is a punctured neighborhood of the point x 0 , on which the function f (x) limited:
.

Let the function have at point x 0 finite non-zero limit:
.
Then, for any number c from the interval , there is such a punctured neighborhood of the point x 0 , what for ,
, If ;
, If .

If, on some punctured neighborhood of the point, , is a constant, then .

If there are finite limits and and on some punctured neighborhood of the point x 0
,
That .

If , and on some neighborhood of the point
,
That .
In particular, if in some neighborhood of a point
,
then if , then and ;
if , then and .

If on some punctured neighborhood of a point x 0 :
,
and there are finite (or infinite of a certain sign) equal limits:
, That
.

Proofs of the main properties are given on the page
"Basic properties of the limits of a function."

Arithmetic properties of the limit of a function

Let the functions and be defined in some punctured neighborhood of the point . And let there be finite limits:
And .
And let C be a constant, that is, a given number. Then
;
;
;
, If .

If, then.

Proofs of arithmetic properties are given on the page
"Arithmetic properties of the limits of a function".

Cauchy criterion for the existence of a limit of a function

Theorem
In order for a function defined on some punctured neighborhood of a finite or at infinity point x 0 , had a finite limit at this point, it is necessary and sufficient that for any ε > 0 there was such a punctured neighborhood of the point x 0 , that for any points and from this neighborhood, the following inequality holds:
.

Limit of a complex function

Theorem on the limit of a complex function
Let the function have a limit and map a punctured neighborhood of a point onto a punctured neighborhood of a point. Let the function be defined on this neighborhood and have a limit on it.
Here are the final or infinitely distant points: . Neighborhoods and their corresponding limits can be either two-sided or one-sided.
Then there is a limit of a complex function and it is equal to:
.

The limit theorem of a complex function is applied when the function is not defined at a point or has a value different from the limit. To apply this theorem, there must be a punctured neighborhood of the point where the set of values ​​of the function does not contain the point:
.

If the function is continuous at point , then the limit sign can be applied to the argument of the continuous function:
.
The following is a theorem corresponding to this case.

Theorem on the limit of a continuous function of a function
Let there be a limit of the function g (t) as t → t 0 , and it is equal to x 0 :
.
Here is point t 0 can be finite or infinitely distant: .
And let the function f (x) is continuous at point x 0 .
Then there is a limit of the complex function f (g(t)), and it is equal to f (x0):
.

Proofs of the theorems are given on the page
"Limit and continuity of a complex function".

Infinitesimal and infinitely large functions

Infinitesimal functions

Definition
A function is said to be infinitesimal if
.

Sum, difference and product of a finite number of infinitesimal functions at is an infinitesimal function at .

Product of a function bounded on some punctured neighborhood of the point , to an infinitesimal at is an infinitesimal function at .

In order for a function to have a finite limit, it is necessary and sufficient that
,
where is an infinitesimal function at .

"Properties of infinitesimal functions".

Infinitely large functions

Definition
A function is said to be infinitely large if
.

The sum or difference of a bounded function, on some punctured neighborhood of the point , and an infinitely large function at is an infinitely large function at .

If the function is infinitely large for , and the function is bounded on some punctured neighborhood of the point , then
.

If the function , on some punctured neighborhood of the point , satisfies the inequality:
,
and the function is infinitesimal at:
, and (on some punctured neighborhood of the point), then
.

Proofs of the properties are presented in section
"Properties of infinitely large functions".

Relationship between infinitely large and infinitesimal functions

From the two previous properties follows the connection between infinitely large and infinitesimal functions.

If a function is infinitely large at , then the function is infinitesimal at .

If a function is infinitesimal for , and , then the function is infinitely large for .

The relationship between an infinitesimal and an infinitely large function can be expressed symbolically:
, .

If an infinitesimal function has a certain sign at , that is, it is positive (or negative) on some punctured neighborhood of the point , then this fact can be expressed as follows:
.
In the same way, if an infinitely large function has a certain sign at , then they write:
.

Then the symbolic connection between infinitely small and infinitely large functions can be supplemented with the following relations:
, ,
, .

Additional formulas relating infinity symbols can be found on the page
"Points at infinity and their properties."

Limits of monotonic functions

Definition
A function defined on some set of real numbers X is called strictly increasing, if for all such that the following inequality holds:
.
Accordingly, for strictly decreasing function the following inequality holds:
.
For non-decreasing:
.
For non-increasing:
.

It follows that a strictly increasing function is also non-decreasing. A strictly decreasing function is also non-increasing.

The function is called monotonous, if it is non-decreasing or non-increasing.

Theorem
Let the function not decrease on the interval where .
If it is bounded above by the number M: then there is a finite limit. If not limited from above, then .
If it is limited from below by the number m: then there is a finite limit. If not limited from below, then .

If points a and b are at infinity, then in the expressions the limit signs mean that .
This theorem can be formulated more compactly.

Let the function not decrease on the interval where . Then there are one-sided limits at points a and b:
;
.

A similar theorem for a non-increasing function.

Let the function not increase on the interval where . Then there are one-sided limits:
;
.

The proof of the theorem is presented on the page
"Limits of monotonic functions".

References:
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.

Today in class we will look at strict sequencing And strict definition of the limit of a function, and also learn to solve relevant problems of a theoretical nature. The article is intended primarily for first-year students of natural sciences and engineering specialties who began to study the theory of mathematical analysis and encountered difficulties in understanding this section of higher mathematics. In addition, the material is quite accessible to high school students.

Over the years of the site’s existence, I have received a dozen letters with approximately the following content: “I don’t understand mathematical analysis well, what should I do?”, “I don’t understand math at all, I’m thinking of quitting my studies,” etc. And indeed, it is the matan who often thins out the student group after the first session. Why is this the case? Because the subject is unimaginably complex? Not at all! The theory of mathematical analysis is not so difficult as it is peculiar. And you need to accept and love her for who she is =)

Let's start with the most difficult case. The first and most important thing is that you don’t have to give up your studies. Understand correctly, you can always quit;-) Of course, if after a year or two you feel sick from your chosen specialty, then yes, you should think about it (and don't get mad!) about a change of activity. But for now it's worth continuing. And please forget the phrase “I don’t understand anything” - it doesn’t happen that you don’t understand anything AT ALL.

What to do if the theory is bad? This, by the way, applies not only to mathematical analysis. If the theory is bad, then first you need to SERIOUSLY focus on practice. In this case, two strategic tasks are solved at once:

– Firstly, a significant share of theoretical knowledge emerged through practice. And that’s why many people understand the theory through... – that’s right! No, no, you're not thinking about that =)

– And, secondly, practical skills will most likely “pull” you through the exam, even if... but let’s not get so excited! Everything is real and everything can be “raised” in a fairly short time. Mathematical analysis is my favorite section of higher mathematics, and therefore I simply could not help but give you a helping hand:

At the beginning of the 1st semester, sequence limits and function limits are usually covered. Don’t understand what these are and don’t know how to solve them? Start with the article Function limits, in which the concept itself is examined “on the fingers” and the simplest examples are analyzed. Next, work through other lessons on the topic, including a lesson about within sequences, on which I have actually already formulated a strict definition.

What symbols besides inequality signs and modulus do you know?

– 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.

Well, is it deadly difficult? =)

After mastering the practice, I look forward to seeing you in the next paragraph:

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

Okay, let's write it down subsequence :

It is not difficult to understand that subsequence approach infinitely close to the number –1, and even-numbered terms – to “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 unique.

Note : the sequence has no limit, but two subsequences can be distinguished from it (see above), each of which has its own limit.

Thus, the above definition turns out to be untenable. Yes, it works for cases like (which I did not 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, except perhaps their final quantities." This is closer to the truth, but still not entirely accurate. So, for example, the sequence half of the terms do not approach zero at all - they are simply equal to it =) By the way, the “flashing light” generally takes two fixed values.

The formulation is not difficult to clarify, but then another question arises: how to write the definition in mathematical symbols? The scientific world struggled with this problem for a long time until the situation was resolved famous maestro, which, in essence, formalized classical mathematical analysis in all its rigor. Cauchy suggested surgery surroundings , which significantly advanced the theory.

Consider some point and its arbitrary-surroundings:

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

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

For example, the “infinite tail” of the sequence will COMPLETELY enter any arbitrarily small neighborhood of the point . So 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 “endless tail” will come in“- members with odd numbers are in fact equal to zero and “do 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.

Let's consolidate the material with practice:

Example 1

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 arbitrary is there any number – such that ALL members 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 that I repeated in class Linear inequalities And Function Domain. In this case, an important circumstance is that “epsilon” and “en” are positive:

Since we are talking about natural numbers on the left, and the right side is generally fractional, 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 inequality and remember what we initially considered arbitrary-neighborhood, i.e. "epsilon" can be equal to anyone a positive number.

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

By the way, from the result obtained a natural pattern is clearly visible: the smaller the neighborhood, the 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 if it is large, however final number of members.

How are your impressions? =) I agree that it’s a bit strange. But strictly! Please re-read and think about everything again.

Let's look at a similar example and get acquainted with other technical techniques:

Example 2

Solution: by definition of a sequence it is necessary to prove that (say it out loud!!!).

Let's consider arbitrary-neighborhood of the point and check, does it exist natural number – such that for all larger numbers the following inequality holds:

To show the existence of such , you need to express “en” through “epsilon”. We simplify the expression under the modulus sign:

The module destroys the minus sign:

The denominator is positive for any “en”, therefore, the sticks can be removed:

Shuffle:

Now we need to extract the square root, but the catch is that for some “epsilon” the right-hand side will be negative. To avoid this trouble let's strengthen inequality by modulus:

Why can this be done? If, relatively speaking, it turns out that , then the condition will also be satisfied. The module can just increase wanted number, and that will suit us too! Roughly speaking, if the hundredth one is suitable, then the two hundredth one is also suitable! According to the definition, you need to show the very fact of the number's existence(at least some), after which all members of the sequence will be in the -neighborhood. By the way, this is why we are not afraid of the final rounding of the right side upward.

Extracting the root:

And round the result:

Conclusion: because the value “epsilon” was chosen arbitrarily, then for any arbitrarily small neighborhood of the point the value was found , such that for all larger numbers the inequality holds . Thus, a-priory. Q.E.D.

I advise especially understanding the strengthening and weakening of inequalities is a typical and very common technique in mathematical analysis. The only thing you need to monitor is the correctness of this or that action. So, for example, inequality under no circumstances is it possible loosen, subtracting, say, one:

Again, conditionally: if the number fits exactly, then the previous one may no longer fit.

The following example for an independent solution:

Example 3

Using the definition of a sequence, prove that

A short solution and answer at the end of the lesson.

If the sequence infinitely large, then the definition of a limit is formulated in a similar way: a point is called a limit of a sequence if for any, as big as you like number, there is a number such that for all larger numbers, the inequality will be satisfied. The number is called vicinity of the point “plus infinity”:

In other words, no matter how large the value we take, the “infinite tail” of the sequence will necessarily go into the -neighborhood of the point, leaving only a finite number of terms on the left.

Standard example:

And shortened notation: , if

For the case, write down the definition yourself. The correct version is at the end of the lesson.

Once you've gotten your head around practical examples and figured out the definition of the limit of a sequence, you can turn to the literature on calculus and/or your lecture notebook. I recommend downloading volume 1 of Bohan (simpler - for correspondence students) and Fichtenholtz (in more detail and detail). Among other authors, I recommend Piskunov, whose course is aimed at technical universities.

Try to conscientiously study the theorems that concern the limit of the sequence, their proofs, consequences. At first, the theory may seem “cloudy”, but this is normal - you just need to get used to it. And many will even get a taste for it!

Rigorous definition of the limit of a function

Let's start with the same thing - how to formulate this concept? The verbal definition of the limit of a function is formulated much simpler: “a number is the limit of a function if with “x” tending to (both left and right), the corresponding function values ​​tend to » (see drawing). Everything seems to be normal, but words are words, meaning is meaning, an icon is an icon, and there are not enough strict mathematical notations. And in the second paragraph we will get acquainted with two approaches to solving this issue.

Let the function be defined on a certain interval, with the possible exception of the point. In educational literature it is generally accepted that the function there Not defined:

This choice emphasizes the essence of the limit of a function: "x" infinitely close approaches , and the corresponding values ​​of the function are infinitely close To . In other words, the concept of a limit does not imply “exact approach” to points, but namely infinitely close approximation, it does not matter whether the function is defined at the point or not.

The first definition of the limit of a function, not surprisingly, is formulated using two sequences. Firstly, the concepts are related, and, secondly, limits of functions are usually studied after limits of sequences.

Consider the sequence points (not on the drawing), belonging to the interval and different from, which converges To . Then the corresponding function values ​​also form a numerical sequence, the members of which are located on the ordinate axis.

Limit of a function according to Heine for any sequences of points (belonging to and different from), which converges to the point , the corresponding sequence of function values ​​converges to .

Eduard Heine is a German mathematician. ...And there is no need to think anything like that, there is only one gay in Europe - Gay-Lussac =)

The second definition of the limit was created... yes, yes, you are right. But first, let's understand its design. Consider an arbitrary -neighborhood of the point (“black” neighborhood). Based on the previous paragraph, the entry means that some value function is located inside the “epsilon” neighborhood.

Now we find the -neighborhood that corresponds to the given -neighborhood (mentally draw black dotted lines from left to right and then from top to bottom). Note that the value is selected along the length of the smaller segment, in this case - along the length of the shorter left segment. Moreover, the “raspberry” -neighborhood of a point can even be reduced, since in the following definition the very fact of existence is important this neighborhood. And, similarly, the notation means that some value is within the “delta” neighborhood.

Cauchy function limit: a number is called the limit of a function at a point if for any pre-selected neighborhood (as small as you like), exists-neighborhood of the point, SUCH, that: AS ONLY values (belonging to) included in this area: (red arrows)– SO IMMEDIATELY the corresponding function values ​​are guaranteed to enter the -neighborhood: (blue arrows).

I must warn you that for the sake of clarity, I improvised a little, so do not overuse =)

Short entry: , if

What is the essence of the definition? Figuratively speaking, by infinitely decreasing the -neighborhood, we “accompany” the function values ​​to their limit, leaving them no alternative to approaching somewhere else. Quite unusual, but again strict! To fully understand the idea, re-read the wording again.

! Attention: if you only need to formulate Heine's definition or just Cauchy definition please don't forget about significant preliminary comments: "Consider a function that is defined on a certain interval, with the possible exception of a point". I stated this once at the very beginning and did not repeat it every time.

According to the corresponding theorem of mathematical analysis, the Heine and Cauchy definitions are equivalent, but the second option is the most famous (still would!), which is also called the "language limit":

Example 4

Using the definition of limit, prove that

Solution: the function is defined on the entire number line except the point. Using the definition, we prove the existence of a limit at a given point.

Note : the value of the “delta” neighborhood depends on the “epsilon”, hence the designation

Let's consider arbitrary-surroundings. The task is to use this value to check whether does it exist-surroundings, SUCH, which from the inequality inequality follows .

Assuming that , we transform the last inequality:
(expanded the quadratic trinomial)