Infinitely small and infinitely large functions. Infinitesimal functions and their basic properties

Function y=f(x) called infinitesimal at x→a or when x→∞, if or , i.e. an infinitesimal function is a function whose limit at a given point is zero.

Examples.

1. Function f(x)=(x-1) 2 is infinitesimal at x→1, since (see figure).

2. Function f(x)= tg x– infinitesimal at x→0.

3. f(x)= log(1+ x) – infinitesimal at x→0.

4. f(x) = 1/x– infinitesimal at x→∞.

Let us establish the following important relationship:

Theorem. If the function y=f(x) representable with x→a as a sum of a constant number b and infinitesimal magnitude α(x): f (x)=b+ α(x) That .

Conversely, if , then f (x)=b+α(x), Where a(x)– infinitesimal at x→a.

Proof.

1. Let us prove the first part of the statement. From equality f(x)=b+α(x) should |f(x) – b|=| α|. But since a(x) is infinitesimal, then for arbitrary ε there is δ – a neighborhood of the point a, in front of everyone x from which, values a(x) satisfy the relation |α(x)|< ε. Then |f(x) – b|< ε. And this means that .

2. If , then for any ε >0 for all X from some δ – neighborhood of a point a will |f(x) – b|< ε. But if we denote f(x) – b= α, That |α(x)|< ε, which means that a– infinitesimal.

Let's consider the basic properties of infinitesimal functions.

Theorem 1. The algebraic sum of two, three, and in general any finite number of infinitesimals is an infinitesimal function.

Proof. Let us give a proof for two terms. Let f(x)=α(x)+β(x), where and . We need to prove that for any arbitrary small ε > 0 found δ> 0, such that for x, satisfying the inequality |x – a|<δ , performed |f(x)|< ε.

So, let’s fix an arbitrary number ε > 0. Since according to the conditions of the theorem α(x) is an infinitesimal function, then there is such δ 1 > 0, which is |x – a|< δ 1 we have |α(x)|< ε / 2. Likewise, since β(x) is infinitesimal, then there is such δ 2 > 0, which is |x – a|< δ 2 we have | β(x)|< ε / 2.

Let's take δ=min(δ 1 , δ2 } .Then in the vicinity of the point a radius δ each of the inequalities will be satisfied |α(x)|< ε / 2 and | β(x)|< ε / 2. Therefore, in this neighborhood there will be

|f(x)|=| α(x)+β(x)| ≤ |α(x)| + | β(x)|< ε /2 + ε /2= ε,

those. |f(x)|< ε, which is what needed to be proved.

Theorem 2. Product of an infinitesimal function a(x) for a limited function f(x) at x→a(or when x→∞) is an infinitesimal function.

Proof. Since the function f(x) is limited, then there is a number M such that for all values x from some neighborhood of a point a|f(x)|≤M. Moreover, since a(x) is an infinitesimal function at x→a, then for an arbitrary ε > 0 there is a neighborhood of the point a, in which the inequality will hold |α(x)|< ε /M. Then in the smaller of these neighborhoods we have | αf|< ε /M= ε. And this means that af– infinitesimal. For the occasion x→∞ the proof is carried out similarly.

From the proven theorem it follows:

Corollary 1. If and, then.

Corollary 2. If c= const, then .

Theorem 3. Ratio of an infinitesimal function α(x) per function f(x), whose limit is different from zero, is an infinitesimal function.

Proof. Let . Then 1 /f(x) there is a limited function. Therefore, a fraction is the product of an infinitesimal function and a limited function, i.e. function is infinitesimal.

Calculus of infinitesimals and larges

Infinitesimal calculus- calculations performed with infinitesimal quantities, in which the derived result is considered as an infinite sum of infinitesimals. The calculus of infinitesimals is a general concept for differential and integral calculus, which forms the basis of modern higher mathematics. The concept of an infinitesimal quantity is closely related to the concept of limit.

Infinitesimal

Subsequence a n called infinitesimal, If . For example, a sequence of numbers is infinitesimal.

The function is called infinitesimal in the vicinity of a point x 0 if .

The function is called infinitesimal at infinity, If or .

Also infinitesimal is a function that is the difference between a function and its limit, that is, if , That f(x) − a = α( x) , .

Infinitely large quantity

In all the formulas below, infinity to the right of equality is implied to have a certain sign (either “plus” or “minus”). That is, for example, the function x sin x, unbounded on both sides, is not infinitely large at .

Subsequence a n called infinitely large, If .

The function is called infinitely large in the vicinity of a point x 0 if .

The function is called infinitely large at infinity, If or .

Properties of infinitely small and infinitely large

Comparison of infinitesimals

How to compare infinitesimal quantities?
The ratio of infinitesimal quantities forms the so-called uncertainty.

Definitions

Suppose we have infinitesimal values α( x) and β( x) (or, which is not important for the definition, infinitesimal sequences).

To calculate such limits it is convenient to use L'Hopital's rule.

Comparison examples

Using ABOUT-symbolism, the results obtained can be written in the following form x 5 = o(x 3). In this case, the following entries are true: 2x 2 + 6x = O(x) And x = O(2x 2 + 6x).

Equivalent values

Definition

If , then the infinitesimal quantities α and β are called equivalent ().
It is obvious that equivalent quantities are a special case of infinitesimal quantities of the same order of smallness.

When the following equivalence relations are valid (as consequences of the so-called remarkable limits):

Theorem

The limit of the quotient (ratio) of two infinitesimal quantities will not change if one of them (or both) is replaced by an equivalent quantity.

This theorem has practical significance when finding limits (see example).

Usage example

Replacing sin 2x equivalent value 2 x, we get

Historical sketch

The concept of “infinitesimal” was discussed back in ancient times in connection with the concept of indivisible atoms, but was not included in classical mathematics. It was revived again with the advent of the “method of indivisibles” in the 16th century - dividing the figure under study into infinitesimal sections.

In the 17th century, the algebraization of infinitesimal calculus took place. They began to be defined as numerical quantities that are less than any finite (non-zero) quantity and yet not equal to zero. The art of analysis consisted in drawing up a relation containing infinitesimals (differentials) and then integrating it.

Old school mathematicians put the concept to the test infinitesimal harsh criticism. Michel Rolle wrote that the new calculus is “ set of ingenious mistakes"; Voltaire caustically remarked that calculus is the art of calculating and accurately measuring things whose existence cannot be proven. Even Huygens admitted that he did not understand the meaning of differentials of higher orders.

As an irony of fate, one can consider the emergence in the middle of the century of non-standard analysis, which proved that the original point of view - actual infinitesimals - was also consistent and could be used as the basis for analysis.

Definition

Subsequence (βn) called an infinitely large sequence, if for any number M, no matter how large, there is a natural number N M depending on M such that for all natural numbers n > N M the inequality holds
|βn | >M.
In this case they write
.
Or at .
They say that it tends to infinity, or converges to infinity.

If, starting from some number N 0 , That
( converges to plus infinity).
If then
( converges to minus infinity).

Let us write these definitions using the logical symbols of existence and universality:
(1) .
(2) .
(3) .

Sequences with limits (2) and (3) are special cases of an infinitely large sequence (1). From these definitions it follows that if the limit of a sequence is equal to plus or minus infinity, then it is also equal to infinity:
.
The reverse, of course, is not true. Members of a sequence may have alternating signs. In this case, the limit can be equal to infinity, but without a specific sign.

Note also that if some property holds for an arbitrary sequence with a limit equal to infinity, then the same property holds for a sequence whose limit is equal to plus or minus infinity.

In many calculus textbooks, the definition of an infinitely large sequence states that the number M is positive: M > 0 . However, this requirement is unnecessary. If it is canceled, then no contradictions arise. It’s just that small or negative values are of no interest to us. We are interested in the behavior of the sequence for arbitrarily large positive values of M. Therefore, if the need arises, then M can be limited from below by any predetermined number a, that is, we can assume that M > a.

When we defined ε - the neighborhood of the end point, then the requirement ε > 0 is an important. For negative values, the inequality cannot be satisfied at all.

Neighborhoods of points at infinity

When we considered finite limits, we introduced the concept of a neighborhood of a point. Recall that a neighborhood of an end point is an open interval containing this point. We can also introduce the concept of neighborhoods of points at infinity.

Let M be an arbitrary number.
Neighborhood of the point "infinity", , is called a set.
Neighborhood of the point "plus infinity", , is called a set.
In the vicinity of the point "minus infinity", , is called a set.

Strictly speaking, the neighborhood of the point "infinity" is the set
(4) ,
where M 1 and M 2 - arbitrary positive numbers. We will use the first definition, since it is simpler. Although, everything said below is also true when using definition (4).

We can now give a unified definition of the limit of a sequence that applies to both finite and infinite limits.

Universal definition of sequence limit.
A point a (finite or at infinity) is the limit of a sequence if for any neighborhood of this point there is a natural number N such that all elements of the sequence with numbers belong to this neighborhood.

Thus, if a limit exists, then outside the neighborhood of point a there can only be a finite number of members of the sequence, or an empty set. This condition is necessary and sufficient. The proof of this property is exactly the same as for finite limits.

Neighborhood property of a convergent sequence
In order for a point a (finite or at infinity) to be a limit of the sequence, it is necessary and sufficient that outside any neighborhood of this point there is a finite number of terms of the sequence or an empty set.
Proof .

Also sometimes the concepts of ε - neighborhoods of points at infinity are introduced.
Recall that the ε-neighborhood of a finite point a is the set .
Let us introduce the following notation. Let ε denote the neighborhood of point a. Then for the end point,
.
For points at infinity:
;
;
.
Using the concepts of ε-neighborhoods, we can give another universal definition of the limit of a sequence:

A point a (finite or at infinity) is the limit of the sequence if for any positive number ε > 0 there is a natural number N ε depending on ε such that for all numbers n > N ε the terms x n belong to the ε-neighborhood of the point a:
.

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

Examples of infinitely large sequences

Example 1

.
Let us write down the definition of an infinitely large sequence:
(1) .
In our case
.

We introduce numbers and , connecting them with inequalities:
.
According to the properties of inequalities, if and , then
.
Note that this inequality holds for any n. Therefore, you can choose like this:
at ;
at .

So, for any one we can find a natural number that satisfies the inequality. Then for everyone,
.
It means that . That is, the sequence is infinitely large.

Example 2

Using the definition of an infinitely large sequence, show that
.

(2) .
The general term of the given sequence has the form:
.

Enter the numbers and:
.
.

Then for any one can find a natural number that satisfies the inequality, so for all ,
.
It means that .

Example 3

Using the definition of an infinitely large sequence, show that
.

Let us write down the definition of the limit of a sequence equal to minus infinity:
(3) .
The general term of the given sequence has the form:
.

Enter the numbers and:
.
From this it is clear that if and , then
.

Since for any one it is possible to find a natural number that satisfies the inequality, then
.

Given , as N we can take any natural number that satisfies the following inequality:
.

Example 4

Using the definition of an infinitely large sequence, show that
.

Let us write down the general term of the sequence:
.
Let us write down the definition of the limit of a sequence equal to plus infinity:
(2) .

Since n is a natural number, n = 1, 2, 3, ... , That
;
;
.

We introduce numbers and M, connecting them with inequalities:
.
From this it is clear that if and , then
.

So, for any number M we can find a natural number that satisfies the inequality. Then for everyone,
.
It means that .

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

See also:

INFINITESMALL FUNCTIONS AND THEIR BASIC PROPERTIES

Function y=f(x) called infinitesimal at x→a or when x→∞, if or , i.e. an infinitesimal function is a function whose limit at a given point is zero.

Examples.

Let us establish the following important relationship:

Theorem. If the function y=f(x) representable with x→a as a sum of a constant number b and infinitesimal magnitude α(x): f (x)=b+ α(x) That .

Conversely, if , then f (x)=b+α(x), Where a(x)– infinitesimal at x→a.

Proof.

Let's consider the basic properties of infinitesimal functions.

Theorem 1. The algebraic sum of two, three, and in general any finite number of infinitesimals is an infinitesimal function.

|f(x)|=| α(x)+β(x)| ≤ |α(x)| + | β(x)|< ε /2 + ε /2= ε,

those. |f(x)|< ε, which is what needed to be proved.

Theorem 2. Product of an infinitesimal function a(x) for a limited function f(x) at x→a(or when x→∞) is an infinitesimal function.

From the proven theorem it follows:

Corollary 1. If and, then.

Corollary 2. If c= const, then .

Theorem 3. Ratio of an infinitesimal function α(x) per function f(x), whose limit is different from zero, is an infinitesimal function.

Proof. Let . Then 1 /f(x) there is a limited function. Therefore the fraction is the product of an infinitesimal function and a bounded function, i.e. function is infinitesimal.

RELATIONSHIP BETWEEN INFINITELY SMALL AND INFINITELY LARGE FUNCTIONS

Theorem 1. If the function f(x) is infinitely large at x→a, then function 1 /f(x) is infinitesimal at x→a.

Proof. Let's take an arbitrary number ε >0 and show that for some δ>0 (depending on ε) for all x, for which |x – a|<δ , the inequality is satisfied, and this will mean that 1/f(x) is an infinitesimal function. Indeed, since f(x) is an infinitely large function at x→a, then there will be δ>0 such that as soon as |x – a|<δ , so | f(x)|> 1/ ε. But then for the same x.

Examples.

The converse theorem can also be proven.

Theorem 2. If the function f(x)- infinitesimal at x→a(or x→∞) and does not vanish, then y= 1/f(x) is an infinitely large function.

Conduct the proof of the theorem yourself.

Examples.

Thus, the simplest properties of infinitesimal and infinitely large functions can be written using the following conditional relations: A≠ 0

LIMIT THEOREMS

Theorem 1. The limit of the algebraic sum of two, three, and generally a certain number of functions is equal to the algebraic sum of the limits of these functions, i.e.

Proof. Let us carry out the proof for two terms, since it can be done in the same way for any number of terms. Let .Then f(x)=b+α(x) And g(x)=c+β(x), Where α And β – infinitesimal functions. Hence,

f(x) + g(x)=(b + c) + (α(x) + β(x)).

Because b+c is a constant, and α(x) + β(x) is an infinitesimal function, then

Example. .

Theorem 2. The limit of the product of two, three, and generally a finite number of functions is equal to the product of the limits of these functions:

Proof. Let . Hence, f(x)=b+α(x) And g(x)=c+β(x) And

fg = (b + α)(c + β) = bc + (bβ + cα + αβ).

Work bc there is a constant value. Function bβ + c α + αβ based on the properties of infinitesimal functions, there is an infinitesimal quantity. That's why .

Corollary 1. The constant factor can be taken beyond the limit sign:

Corollary 2. The degree limit is equal to the limit degree:

Example..

Theorem 3. The limit of the quotient of two functions is equal to the quotient of the limits of these functions if the limit of the denominator is different from zero, i.e.

Proof. Let . Hence, f(x)=b+α(x) And g(x)=c+β(x), Where α, β – infinitesimal. Let's consider the quotient

A fraction is an infinitesimal function because the numerator is an infinitesimal function and the denominator has a limit c 2 ≠0.

Examples.

Theorem 4. Let three functions be given f(x), u(x) And v(x), satisfying the inequalities u (x)≤f(x)≤ v(x). If the functions u(x) And v(x) have the same limit at x→a(or x→∞), then the function f(x) tends to the same limit, i.e. If

, That .

The meaning of this theorem is clear from the figure.

The proof of Theorem 4 can be found, for example, in the textbook: Piskunov N. S. Differential and integral calculus, vol. 1 - M.: Nauka, 1985.

Theorem 5. If at x→a(or x→∞) function y=f(x) accepts non-negative values y≥0 and at the same time tends to the limit b, then this limit cannot be negative: b≥0.

Proof. We will carry out the proof by contradiction. Let's pretend that b<0 , Then |y – b|≥|b| and, therefore, the difference modulus does not tend to zero when x→a. But then y does not reach the limit b at x→a, which contradicts the conditions of the theorem.

Theorem 6. If two functions f(x) And g(x) for all values of the argument x satisfy the inequality f(x)≥ g(x) and have limits, then the inequality holds b≥c.

Proof. According to the conditions of the theorem f(x)-g(x) ≥0, therefore, by Theorem 5 , or .

UNILATERAL LIMITS

So far we have considered determining the limit of a function when x→a in an arbitrary manner, i.e. the limit of the function did not depend on how it was located x towards a, to the left or right of a. However, it is quite common to find functions that have no limit under this condition, but they do have a limit if x→a, remaining on one side of A, left or right (see figure). Therefore, the concepts of one-sided limits are introduced.

If f(x) tends to the limit b at x tending to a certain number a So x accepts only values less than a, then they write and call blimit of the function f(x) at point a on the left.

Definitions and properties of infinitesimal and infinitely large functions at a point. Proofs of properties and theorems. Relationship between infinitesimal and infinitely large functions.

Content

See also: Infinitesimal sequences - definition and properties
Properties of infinitely large sequences

Definition of infinitesimal and infinitesimal functions

Let x 0 is a finite or infinite point: ∞, -∞ or +∞.

Definition of an infinitesimal function
Function α (x) called infinitesimal as x tends to x 0 0 , and it is equal to zero:
.

Definition of an infinitely large function
Function f (x) called infinitely large as x tends to x 0 , if the function has a limit as x → x 0 , and it is equal to infinity:
.

Properties of infinitesimal functions

Property of the sum, difference and product of infinitesimal functions

Sum, difference and product finite number of infinitesimal functions as x → x 0 is an infinitesimal function as x → x 0 .

This property is a direct consequence of the arithmetic properties of the limits of a function.

Theorem on the product of a bounded function and an infinitesimal

Product of a function bounded on some punctured neighborhood of point x 0 , to infinitesimal, as x → x 0 , is an infinitesimal function as x → x 0 .

The property of representing a function as the sum of a constant and an infinitesimal function

In order for the function f (x) had a finite limit, it is necessary and sufficient that
,
where is an infinitesimal function as x → x 0 .

Properties of infinitely large functions

Theorem on the sum of a bounded function and an infinitely large

The sum or difference of a bounded function on some punctured neighborhood of the point x 0 , and an infinitely large function, as x → x 0 , is an infinitely large function as x → x 0 .

Theorem on the division of a bounded function by an infinitely large one

If function f (x) is infinitely large as x → x 0 , and the function g (x)- is bounded on some punctured neighborhood of point x 0 , That
.

Theorem on the division of a function bounded below by an infinitesimal one

If the function , on some punctured neighborhood of the point , is bounded from below by a positive number in absolute value:
,
and the function is infinitesimal as x → x 0 :
,
and there is a punctured neighborhood of the point on which , then
.

Property of inequalities of infinitely large functions

If the function is infinitely large at:
,
and the functions and , on some punctured neighborhood of the point satisfy the inequality:
,
then the function is also infinitely large at:
.

This property has two special cases.

Let, on some punctured neighborhood of the point , the functions and satisfy the inequality:
.
Then if , then and .
If , then and .

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 we can write it like this:
.
In the same way, if an infinitely large function has a certain sign at , then they write:
, or .

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."

Proof of properties and theorems

Proof of the theorem on the product of a bounded function and an infinitesimal one

To prove this theorem, we will use . We also use the property of infinitesimal sequences, according to which

Let the function be infinitesimal at , and let the function be bounded in some punctured neighborhood of the point:
at .

Since there is a limit, there is a punctured neighborhood of the point on which the function is defined. Let there be an intersection of neighborhoods and . Then the functions and are defined on it.

.
,
a sequence is infinitesimal:
.

Let us take advantage of the fact that the product of a bounded sequence and an infinitesimal sequence is an infinitesimal sequence:
.
.

The theorem has been proven.

Proof of the property of representing a function as the sum of a constant and an infinitesimal function

Necessity. Let the function have a finite limit at a point
.
Consider the function:
.
Using the property of the limit of the difference of functions, we have:
.
That is, there is an infinitesimal function at .

Adequacy. Let it be. Let's apply the property of the limit of the sum of functions:
.

The property has been proven.

Proof of the theorem on the sum of a bounded function and an infinitely large

To prove the theorem, we will use Heine’s definition of the limit of a function

at .

Since there is a limit, there is a punctured neighborhood of the point on which the function is defined. Let there be an intersection of neighborhoods and . Then the functions and are defined on it.

Let there be an arbitrary sequence converging to , whose elements belong to the neighborhood:
.
Then the sequences and are defined. Moreover, the sequence is limited:
,
a sequence is infinitely large:
.

Since the sum or difference of a limited sequence and an infinitely large
.
Then, according to the definition of the limit of a sequence according to Heine,
.

The theorem has been proven.

Proof of the theorem on the quotient of division of a bounded function by an infinitely large one

To prove this, we will use Heine's definition of the limit of a function. We also use the property of infinitely large sequences, according to which is an infinitesimal sequence.

Let the function be infinitely large at , and let the function be bounded in some punctured neighborhood of the point:
at .

Since the function is infinitely large, there is a punctured neighborhood of the point where it is defined and does not vanish:
at .
Let there be an intersection of neighborhoods and . Then the functions and are defined on it.

Since the quotient of dividing a limited sequence by an infinitely large one is an infinitesimal sequence, then
.
Then, according to the definition of the limit of a sequence according to Heine,
.

The theorem has been proven.

Proof of the quotient theorem for dividing a function bounded below by an infinitesimal one

To prove this property, we will use Heine's definition of the limit of a function. We also use the property of infinitely large sequences, according to which is an infinitely large sequence.

Let the function be infinitesimal for , and let the function be bounded in absolute value from below by a positive number, on some punctured neighborhood of the point:
at .

By condition, there is a punctured neighborhood of the point on which the function is defined and does not vanish:
at .
Let there be an intersection of neighborhoods and . Then the functions and are defined on it. Moreover, and .

Let there be an arbitrary sequence converging to , whose elements belong to the neighborhood:
.
Then the sequences and are defined. Moreover, the sequence is bounded below:
,
and the sequence is infinitesimal with non-zero terms:
, .

Since the quotient of dividing a sequence bounded below by an infinitesimal one is an infinitely large sequence, then
.
And let there be a punctured neighborhood of the point on which
at .

Let us take an arbitrary sequence converging to . Then, starting from some number N, the elements of the sequence will belong to this neighborhood:
at .
Then
at .

According to the definition of the limit of a function according to Heine,
.
Then, by the property of inequalities of infinitely large sequences,
.
Since the sequence is arbitrary, converging to , then, by the definition of the limit of a function according to Heine,
.

The property has been proven.

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

Infinitely small and infinitely large functions. Infinitesimal functions and their basic properties

Calculus of infinitesimals and larges

Infinitesimal

Infinitely large quantity

Properties of infinitely small and infinitely large

Comparison of infinitesimals

Definitions

Comparison examples

Equivalent values

Definition

Theorem

Usage example

Historical sketch

see also

See what “Infinitesimal quantity” is in other dictionaries:

Definition

Neighborhoods of points at infinity

Examples of infinitely large sequences

Example 1

Example 2

Example 3

Example 4

Definition of infinitesimal and infinitesimal functions

Properties of infinitesimal functions

Properties of infinitely large functions

Relationship between infinitely large and infinitesimal functions

Proof of properties and theorems

Proof of the theorem on the product of a bounded function and an infinitesimal one

Proof of the property of representing a function as the sum of a constant and an infinitesimal function

Proof of the theorem on the sum of a bounded function and an infinitely large

Proof of the theorem on the quotient of division of a bounded function by an infinitely large one

Proof of the quotient theorem for dividing a function bounded below by an infinitesimal one