Definition of an infinitely large function. Definition of an infinitely large sequence: Infinitely large and their properties

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

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 a 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 a 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.

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 with nonzero terms:
, .

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.

Infinitesimal functions

The function %%f(x)%% is called infinitesimal(b.m.) with %%x \to a \in \overline(\mathbb(R))%%, if with this tendency of the argument the limit of the function is equal to zero.

The concept of b.m. function is inextricably linked with instructions to change its argument. We can talk about b.m. functions at %%a \to a + 0%% and at %%a \to a - 0%%. Usually b.m. functions are denoted by the first letters of the Greek alphabet %%\alpha, \beta, \gamma, \ldots%%

Examples

1. The function %%f(x) = x%% is b.m. at %%x \to 0%%, since its limit at the point %%a = 0%% is zero. According to the theorem about the connection between the two-sided limit and the one-sided limit, this function is b.m. both with %%x \to +0%% and with %%x \to -0%%.
2. Function %%f(x) = 1/(x^2)%% - b.m. at %%x \to \infty%% (as well as at %%x \to +\infty%% and at %%x \to -\infty%%).

A non-zero constant number, no matter how small in absolute value, is not a b.m. function. For constant numbers, the only exception is zero, since the function %%f(x) \equiv 0%% has a zero limit.

Theorem

The function %%f(x)%% has at the point %%a \in \overline(\mathbb(R))%% of the extended number line a final limit equal to the number %%b%% if and only if this function equal to the sum of this number %%b%% and b.m. functions %%\alpha(x)%% with %%x \to a%%, or $$ \exists~\lim\limits_(x \to a)(f(x)) = b \in \mathbb(R ) \Leftrightarrow \left(f(x) = b + \alpha(x)\right) \land \left(\lim\limits_(x \to a)(\alpha(x) = 0)\right). $$

Properties of infinitesimal functions

According to the rules of passage to the limit with %%c_k = 1~ \forall k = \overline(1, m), m \in \mathbb(N)%%, the following statements follow:

1. The sum of the final number of b.m. functions for %%x \to a%% is b.m. at %%x \to a%%.
2. The product of any number b.m. functions for %%x \to a%% is b.m. at %%x \to a%%.

Product b.m. functions at %%x \to a%% and a function bounded in some punctured neighborhood %%\stackrel(\circ)(\text(U))(a)%% of point a, there is b.m. at %%x \to a%% function.

It is clear that the product of a constant function and b.m. at %%x \to a%% there is b.m. function at %%x \to a%%.

Equivalent infinitesimal functions

Infinitesimal functions %%\alpha(x), \beta(x)%% for %%x \to a%% are called equivalent and write %%\alpha(x) \sim \beta(x)%%, if

$$ \lim\limits_(x \to a)(\frac(\alpha(x))(\beta(x))) = \lim\limits_(x \to a)(\frac(\beta(x) )(\alpha(x))) = 1. $$

Theorem on the replacement of b.m. functions equivalent

Let %%\alpha(x), \alpha_1(x), \beta(x), \beta_1(x)%% be b.m. functions for %%x \to a%%, with %%\alpha(x) \sim \alpha_1(x); \beta(x) \sim \beta_1(x)%%, then $$ \lim\limits_(x \to a)(\frac(\alpha(x))(\beta(x))) = \lim\ limits_(x \to a)(\frac(\alpha_1(x))(\beta_1(x))). $$

Equivalent b.m. functions.

Let %%\alpha(x)%% be b.m. function at %%x \to a%%, then

1. %%\sin(\alpha(x)) \sim \alpha(x)%%
2. %%\displaystyle 1 - \cos(\alpha(x)) \sim \frac(\alpha^2(x))(2)%%
3. %%\tan \alpha(x) \sim \alpha(x)%%
4. %%\arcsin\alpha(x) \sim \alpha(x)%%
5. %%\arctan\alpha(x) \sim \alpha(x)%%
6. %%\ln(1 + \alpha(x)) \sim \alpha(x)%%
7. %%\displaystyle\sqrt[n](1 + \alpha(x)) - 1 \sim \frac(\alpha(x))(n)%%
8. %%\displaystyle a^(\alpha(x)) - 1 \sim \alpha(x) \ln(a)%%

Example

$$ \begin(array)(ll) \lim\limits_(x \to 0)( \frac(\ln\cos x)(\sqrt(1 + x^2) - 1)) & = \lim\limits_ (x \to 0)(\frac(\ln(1 + (\cos x - 1)))(\frac(x^2)(4))) = \\ & = \lim\limits_(x \to 0)(\frac(4(\cos x - 1))(x^2)) = \\ & = \lim\limits_(x \to 0)(-\frac(4 x^2)(2 x^ 2)) = -2 \end(array) $$

Infinitely large functions

The function %%f(x)%% is called infinitely large(b.b.) with %%x \to a \in \overline(\mathbb(R))%%, if with this tendency of the argument the function has an infinite limit.

Similar to b.m. functions concept b.b. function is inextricably linked with instructions to change its argument. We can talk about b.b. functions for %%x \to a + 0%% and %%x \to a - 0%%. The term “infinitely large” does not speak about the absolute value of the function, but about the nature of its change in the vicinity of the point in question. No constant number, no matter how large in absolute value, is infinitely large.

Examples

1. Function %%f(x) = 1/x%% - b.b. at %%x \to 0%%.
2. Function %%f(x) = x%% - b.b. at %%x \to \infty%%.

If the definition conditions $$ \begin(array)(l) \lim\limits_(x \to a)(f(x)) = +\infty, \\ \lim\limits_(x \to a)(f( x)) = -\infty, \end(array) $$

then they talk about positive or negative b.b. at %%a%% function.

Example

Function %%1/(x^2)%% - positive b.b. at %%x \to 0%%.

The connection between b.b. and b.m. functions

If %%f(x)%% is b.b. with %%x \to a%% function, then %%1/f(x)%% - b.m.

at %%x \to a%%. If %%\alpha(x)%% - b.m. for %%x \to a%% is a non-zero function in some punctured neighborhood of the point %%a%%, then %%1/\alpha(x)%% is b.b. at %%x \to a%%.

Properties of infinitely large functions

Let us present several properties of the b.b. functions. These properties follow directly from the definition of b.b. functions and properties of functions having finite limits, as well as from the theorem on the connection between b.b. and b.m. functions.

1. The product of a finite number of b.b. functions for %%x \to a%% is b.b. function at %%x \to a%%. Indeed, if %%f_k(x), k = \overline(1, n)%% - b.b. function at %%x \to a%%, then in some punctured neighborhood of the point %%a%% %%f_k(x) \ne 0%%, and by the connection theorem b.b. and b.m. functions %%1/f_k(x)%% - b.m. function at %%x \to a%%. It turns out %%\displaystyle\prod^(n)_(k = 1) 1/f_k(x)%% - b.m function for %%x \to a%%, and %%\displaystyle\prod^(n )_(k = 1)f_k(x)%% - b.b. function at %%x \to a%%.
2. Product b.b. functions for %%x \to a%% and a function which in some punctured neighborhood of the point %%a%% in absolute value is greater than a positive constant is b.b. function at %%x \to a%%. In particular, the product b.b. a function with %%x \to a%% and a function that has a finite non-zero limit at the point %%a%% will be b.b. function at %%x \to a%%.

The sum of a function bounded in some punctured neighborhood of the point %%a%% and b.b. functions with %%x \to a%% is b.b. function at %%x \to a%%.

For example, the functions %%x - \sin x%% and %%x + \cos x%% are b.b. at %%x \to \infty%%.

3. The sum of two b.b. functions at %%x \to a%% there is uncertainty. Depending on the sign of the terms, the nature of the change in such a sum can be very different.

   Example

   Let the functions %%f(x)= x, g(x) = 2x, h(x) = -x, v(x) = x + \sin x%% be given. functions at %%x \to \infty%%. Then:

   • %%f(x) + g(x) = 3x%% - b.b. function at %%x \to \infty%%;
   • %%f(x) + h(x) = 0%% - b.m. function at %%x \to \infty%%;
   • %%h(x) + v(x) = \sin x%% has no limit at %%x \to \infty%%.

Definition of a numerical function. Methods for specifying functions.

Let D be a set on the number line R. If each x belonging to D is associated with a single number y=f(x), then we say that a function f is given.

Methods for specifying functions:

1) tabular – for functions defined on a finite set.

2) analytical

3) graphic

2 and 3 – for functions defined on an infinite set.

The concept of an inverse function.

If the function y=f(x) is such that different values ​​of the x argument correspond to different values ​​of the function, then the variable x can be expressed as a function of the variable y: x=g(y). The function g is called the inverse of f and is denoted by f^(-1).

The concept of a complex function.

A complex function is a function whose argument is any other function.

Let functions f(x) and g(x) be given. Let's make two complex functions out of them. Considering the function f to be external (main), and the function g to be internal, we obtain a complex function u(x)=f(g(x)).

Determination of the sequence limit.

A number a is called the limit of a sequence (xn) if for any positive there is a number n0, starting from which all terms of the sequence differ from a in modulus by less than ε (i.e., they fall into the ε-neighborhood of the point a):

Rules for calculating the limits of convergent sequences.

1. Every convergent sequence has only one limit. 2. If all elements of the sequence (x n) are equal to C (constant), then the limit of the sequence (x n) is also equal to C. 3. ; 4. ; 5. .

Definition of a limited sequence.

The sequence (x n) is called bounded if the set of numbers X=(x n) is bounded: .

Definition of an infinitesimal sequence.

A sequence (x n) is said to be infinitesimal if for any (no matter how small) >0 there is a number n 0 such that for any n>n 0 the inequality |x n |< .

Definition of an infinitely large sequence.

A sequence is said to be infinitely large if for any (no matter how large) number A>0 there is a number n 0 such that for every number n>n 0 the inequality |x n |>A holds.

Definition of monotonic sequences.

Monotonous sequences: 1) increasing ifx n x n +1 for all n, 4) non-increasing if x n x n +1 for all n.

Determination of the limit of a function at a point.

The limit of the function y=f(x) at the point x 0 (or at x x 0) is the number a if for any sequence (x n) values ​​of the argument converging to x 0 (all x n x 0), The sequence of (f(x n)) values ​​of the function converges to the limit a.

Definition of an infinitesimal function.

F-iya f(x) is said to be infinitesimal as x→A if .

Definition of an infinitely large function.

F-iya f(x) is said to be infinitely large for x→A if .

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

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 .

In all cases, 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 is not infinitely large at .

Properties of infinitely small and infinitely large

Comparison of infinitesimal quantities

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

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: , , .

Theorem

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

Usage example

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.

see also

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See what “Infinitely large” is in other dictionaries:

The variable quantity Y is the inverse of the infinitesimal quantity X, that is, Y = 1/X... Big Encyclopedic Dictionary

The variable y is the inverse of the infinitesimal x, that is, y = 1/x. * * * INFINITELY LARGE INFINITELY LARGE, variable quantity Y, inverse to the infinitesimal quantity X, that is, Y = 1/X ... encyclopedic Dictionary

In mathematics, a variable quantity that, in a given process of change, becomes and remains greater in absolute value than any predetermined number. Study of B. b. quantities can be reduced to the study of infinitesimals (See... ... Great Soviet Encyclopedia

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 arbitrary arbitrarily 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), the limit of which 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.