Y tgx is an infinitely large function at. Definition of an infinitely large sequence

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

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: , , .

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.

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

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 constant number other than zero, 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%%, and %%\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. functions 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%%.

The definition of an infinitely large sequence is given. The concepts of neighborhoods of points at infinity are considered. A universal definition of the limit of a sequence is given, which applies to both finite and infinite limits. Examples of application of the definition of an infinitely large sequence are considered.

Content

See also: Determining the Sequence Limit

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
|β n | >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 can 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 the 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:

Def: The function is called infinitesimal at , if .

In the notation “ ” we will assume that x 0 can take as final value: x 0= Сonst, and infinite: x 0= ∞.

Properties of infinitesimal functions:

1) The algebraic sum of a finite number of infinitesimal functions is an infinitesimal sum of functions.

2) The product of a finite number of infinitesimal functions is an infinitesimal function.

3) The product of a bounded function and an infinitesimal function is an infinitesimal function.

4) The quotient of dividing an infinitesimal function by a function whose limit is nonzero is an infinitesimal function.

Example: Function y = 2 + x is infinitesimal at , because .

Def: The function is called infinitely large at , if .

Properties of infinitely large functions:

1) The sum of infinitely large functions is an infinitely large function.

2) The product of an infinitely large function and a function whose limit is nonzero is an infinitely large function.

3) The sum of an infinitely large function and a bounded function is an infinitely large function.

4) The quotient of dividing an infinitely large function by a function that has a finite limit is an infinitely large function.

Example: Function y= is infinitely large at , because .

Theorem.Relationship between infinitely small and infinitely large quantities. If a function is infinitesimal at , then the function is infinitely large at . And conversely, if a function is infinitely large at , then the function is infinitesimal at .

The ratio of two infinitesimals is usually denoted by the symbol, and the ratio of two infinitesimals by the symbol. Both relations are indefinite in the sense that their limit may or may not exist, be equal to a certain number or be infinite, depending on the type of specific functions included in the indefinite expressions.

In addition to uncertainties of type and uncertainties, the following expressions are:

Difference of infinitely large ones of the same sign;

The product of an infinitesimal by an infinitely large;

An exponential function whose base tends to 1 and exponent tends to ;

An exponential function whose base is infinitesimal and whose exponent is infinitely large;

An exponential function whose base and exponent are infinitesimal;

An exponential function whose base is infinitely large and whose exponent is infinitesimal.

It is said that there is uncertainty of the corresponding type. The limit calculation is called in these cases revealing uncertainty. To reveal uncertainty, the expression under the limit sign is converted to a form that does not contain uncertainty.

When calculating limits, the properties of limits are used, as well as the properties of infinitesimal and infinitely large functions.

Let's look at examples of calculations of various limits.

1) . 2) .

4) , because product of an infinitesimal function at and a bounded function is infinitesimal.

5) . 6) .

7) = =

. In this case, there was an uncertainty of type, which was revealed by factoring the polynomials and reducing them to a common factor.

= .

In this case, there was an uncertainty of type , which was resolved by multiplying the numerator and denominator by the expression, using the formula, and then reducing the fraction by (+1).

9)
. In this example, type uncertainty was revealed by dividing the numerator and denominator of the fraction by the leading power.

Wonderful Limits

The first wonderful limit : .

Proof. Let's consider the unit circle (Fig. 3).

Fig.3. Unit circle

Let X– radian measure of the central angle MOA(), Then OA = R= 1, MK= sin x, AT= tg x. Comparing the areas of triangles OMA, OTA and sectors OMA, we get:

,

.

Divide the last inequality by sin x, we get:

.

Since at , then by property 5) limits

Hence the inverse value for , which is what needed to be proved.

Comment: If the function is infinitesimal at , i.e. , then the first remarkable limit has the form:

.

Let's look at examples of limit calculations using the first remarkable limit.

When calculating this limit, we used the trigonometric formula: .

.

Let's look at examples of limit calculations using the second remarkable limit.

2) .

3) . There is type uncertainty. Let's make a replacement, then; at .

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 a 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.