The same order of smallness. Examples. Proof Properties on the presentation of a function in the form of a permanent and infinitely small function

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The concept of infinitely small and infinitely large values \u200b\u200bplays an important role in mathematical analysis. Many tasks are simply and easily solved using the concepts of infinitely large and small values.

Infinitely small.

The variable is called infinitely small if for any value to enforce it that each following it is denied the absolute value less.

If a - infinitely smallthey say that she strives for zero, and write:.

Infinitely large.

Variable x. called infinitely bigif for any positive number c.there is such a value that every following x. It will be more absolute more. Write:

The quantity infinitely big, there is a value infinitely small, and back.

10. Properties of the function limits

1) Limit of constant value

The constant value limit is equal to the most constant value:

2) The limit of the amount

The limit of the amount of two functions is equal to the sum of the limits of these functions:

Similarly, the difference limit of two functions is equal to the difference between these functions.

Extended amount of the amount of the amount:

The limit of the sum of several functions is equal to the sum of the limits of these functions:

Similarly, the limit of the difference between several functions is equal to the difference between these functions.

3) The limit of the function for a permanent value

Permanent coefficient can be made for the limit:

4) Limit of the work

The limit of the product of two functions is equal to the limits of these functions:

Extended Product Limit Property

The limit of the product of several functions is equal to the limits of these functions:

5) Private limit

The limit of the private two functions is equal to the attitude of the limits of these functions, provided that the denominator limit is not zero:

11. First wonderful limit

Evidence

Consider one-sided limits and prove that they are equal to 1.

Let be . I will postpone this angle on a single circle ().

Point K. - Point of intersection of the beam with a circle and a point L. - With a tangent of a single circle at the point. Point H. - Projection Point K. on the axis OX..

It's obvious that:

(Where - Sector Square)

Substituting in (1), we get:

Since when:

We multiply on:

Let us turn to the limit:

Find the left one-sided limit:

The right and left one-sided limits exist and are equal to 1, which means the limit itself is 1.

12-13. The second wonderful limit

Proof of the second remarkable limit:

Knowing that the second wonderful limit is faithful for natural values \u200b\u200bx, we will prove the second wonderful limit for real X, that is, we prove that . Consider two cases:

1. Let. Each value of X is concluded between two positive integers: where is a whole part x.

From here it follows: therefore

If, then. Therefore, according to the limit We have:

In the sign (about the limit of the intermediate function) of the existence of the limits .

2. Let. Make a substitution, then

Of these two cases, it follows that for real X.

14. Private derivatives.

Let be z \u003d F.(x, Y.) . Fix any point (x, Y.), and then without changing the fixed value of the argument y., give the argument x. increment. Then z. will receive an increment called private increment z. by x. and denotes and determined by the formula.

Similarly, if x. preserves a constant value, and y. gets increment then z. Gets a private increment z. by y.,.

Definition. Private derived by x. from function z \u003d F.(x, Y.) called the limit of the relationship of private increment x. to increment when desire for zero, i.e.

The private derivative is indicated by one of the characters .

Similarly, a private derivative is determined by y.:

Thus, the private derivatives of the two variables are calculated according to the same rules as the derivatives of one variable.

Example. Find private derived functions z \u003d X. 2 e. x-2y .

Private derivatives of any number of variables are defined similarly.

Comparison of infinitely small functions, equivalent functions

Infinitely small and infinitely large values.

O.1. The sequence is called infinitely bigif for any positive number A (how much we would not have taken it) There is no number n such that at n\u003e n inequality is performed | x P | \u003e And those. Whatever a large number and we have taken, there is such a number, starting from which all members of the sequence will be more than A.

Definition 6.. The sequence (α n) is called infinitely smallif for any positive number ε (how small we would not have taken it) there is a number n such that at n\u003e n inequality is performed | α P | \u003cΕ.

1. The sequence (P) is infinitely large.

2. The sequence () is infinitely small.

Theorem 1. If (x n) is an infinitely large sequence and all its members are different from zero, x n ≠ 0, then the sequence (α n) \u003d is infinitely small, and, back, if (α n) is infinitely small sequence, α n ≠ 0 , then the sequence (x n) \u003d infinitely large.

We formulate the basic properties of infinitely small sequences in the form of the theorems.

Theorem 2. The amount and difference of two infinitely small sequences are infinitely small sequences.

Example 2.The sequence with a shared member is infinitely small, because Those, the specified sequence is the sum of infinitely small sequences and therefore is infinitely small.

Corollary. The algebraic amount of any finite number of infinitely small sequences is an infinitely small sequence.

Theorem 3. The work of two infinitely small sequences is an infinitely small sequence.

Corollary. The product of any finite number of infinitely small sequences is an infinitely small sequence.

Comment. Private two infinitely small sequences can be any sequence and may not make sense.

For example, if,, all the elements of the sequence are equal to 1 and this sequence is limited. If, then the sequence is infinitely large, and vice versa, if, and, then the infinitely small sequence. If, starting from some number, the sequence elements are zero, then the sequence does not make sense.

Theorem 4. The product of a limited sequence is infinitely small there is an infinitely small sequence.

Example 3. The sequence is infinitely small, because And the sequence () is infinitely small, the sequence is limited, because \u003c1. Consequently, the infinitely small sequence.

Corollary. The product of an infinitely small sequence on the number is an infinitely small sequence.

Definition. Function F (x) is called infinitely big If, for any, even how much of a large positive number, there is such a positive number (depending on M, δ \u003d δ (M)), which for all x, not equal to x 0 and satisfying the condition, inequality is carried out

Record: or when.

For example, the function is an infinitely large function at; Function at.

If F (x) tends to infinity when and takes only positive values, then they write, if only negative values, then.

Definition. Function f (x) specified on the entire numeric line called infinitely big If, for any positive number, there is such a positive number (depending on m, n \u003d n (m)), that at all, satisfying the condition, inequality is performed

For example, the function y \u003d 2 x is an infinitely large function at; The function is an infinitely large function at.

Properties of infinitely large functions:

1. Production B.B.F. On a function whose limit is different from zero, there is a B.B.F.

2. Amount B.B.F. and limited function is B.B.F.

3. Private from division B.B.F. The function having a limit is B.B.F.

For example, if the function f (x) \u003d TGX is B.B.F. when, the function φ (x) \u003d 4x-3, when it has a limit (2π-3), different from zero, and the function ψ (x) \u003d sinx is a limited function, then

f (x) φ (x) \u003d (4x-3) tgx; f (x) + ψ (x) \u003d TGX + SINX; There are infinitely large functions at.

Definition. Function F (x) is called infinitely smallwhen, if

By defining the limit of the function, equality (1) means: for any, even an arbitrarily small positive number, there is such a positive number (depending on ε, δ \u003d δ (ε)), which for all x, not equal to x 0 and satisfying the condition, Inequality is performed

Theorem. To perform equality it is necessary and enough for the function to be infinitely small at. In this case, the function can be represented in the form.

Similarly determined by B.M.F. at, - 0,, in all cases f (x) 0.

Infinitely small functions are often called infinitely small values \u200b\u200bor infinitely small; Denote by the Greek letters α, β, etc.

For example, y \u003d x 2 at x → 0; y \u003d x-2 at x → 2; y \u003d sinx at x → πk, - infinitely small functions.

Properties of infinitely small functions:

1. The sum of the final number of infinitely small functions is the magnitude of infinitely small;

2. The product of a finite number of infinitely small functions, as well as infinitely small functions on a limited function, there is a magnitude infinitely small;

3. Private from division of infinitely small functions on a function whose limit is not equal to no longer if the magnitude is infinitely small.

Consider the last property if the functions are infinitely small (comparison of infinitely small functions):

one). If, then called an infinitely small, higher order of smallness than.

Example. At X → 2, the function (x - 2) 3 is infinitely small higher than (x -2), since.

2). If, then they are called infinitely small one order (they have the same speed of desire for nulo);

Example. At x → 0, the functions 5x 2 and x 2 are infinitely small one orders, as.

3). If, then are called equivalent infinitely small, designated ~., Then

The relationship between infinitely small and infinitely large functions: the function is opposite infinitely small is infinitely large (and vice versa), i.e. If - an infinitely small function, then - infinitely big.

Calculus infinitely small and large

The calculus is infinitely small - The calculations produced with infinitely small values \u200b\u200bin which the derivative result is considered as an infinite amount of infinitely small. The calculation of infinitely small values \u200b\u200bis a general concept for differential and integral calculations that make up the basis of modern higher mathematics. The concept of infinitely small magnitude is closely related to the concept of limit.

Infinitely small

Sequence a. n. called infinitely small, if a . For example, the sequence of numbers is infinitely small.

The function is called infinitely small in the neighborhood x. 0, if .

The function is called infinitely small on infinity, if a or .

The function is also infinitely small, representing the difference between the function and its limit, that is, if T. f.(x.) − a. = α( x.) , .

Infinitely large amount

In all the formulas below, the infinity of the right of equality is meant a certain mark (or "plus" or "minus"). That is, for example, a function x.sin. x. Unlimited on both sides is not infinitely large at.

Sequence a. n. called infinitely big, if a .

The function is called infinitely large in the neighborhood x. 0, if .

The function is called infinitely big on infinity, if a or .

Properties of infinitely small and infinitely large

Comparison of infinitely small values

How to compare infinitely small values?
The ratio of infinitely small values \u200b\u200bforms the so-called uncertainty.

Definitions

Suppose we have infinitely small at the same size α ( x.) and β ( x.) (either, which is not important for determining, infinitely small sequences).

To calculate such limits, it is convenient to use the Lopital rule.

Examples of comparison

Using ABOUT-Simvoliki obtained results can be recorded in the following form x. 5 = o.(x. 3). In this case, the records are valid 2x. 2 + 6x. = O.(x.) and x. = O.(2x. 2 + 6x.).

Equivalent quantities

Definition

If, the infinitely small values \u200b\u200bof α and β are called equivalent ().
Obviously, equivalent values \u200b\u200bare a special case of infinitely small values \u200b\u200bof one order of smallness.

If the following equivalence ratios are true (as a result of the so-called. Wonderful limits):

Theorem

The limit of the private (relationship) of two infinitely small values \u200b\u200bwill not change, if one of them (or both) replace the equivalent value.

This theorem has an applied value when finding limits (see example).

Example of use

Replacing s.i.n.2x. Equivalent of 2. x. Receive

Historical essay

The concept of "infinitely small" was discussed in ancient times due to the concept of indivisible atoms, but it was not included in the classical mathematics. It was again revived with the advent of the "method of indivisible" in the XVI century - the division of the studied figure on infinitely small sections.

In the XVII century, algebraization of calculus is infinitely small. They began to be defined as numeric values \u200b\u200bthat are less than any finite (nonzero) value and are not equal to zero. The art of analysis was to compile the relation containing infinitely small (differentials), and then in its integration.

Mathematics of the old school subjected the concept infinitely small sharp criticism. Michelle roll wrote that there is a new calculus " a set of ingenious errors"; Voltaire poisonously noticed that this calculus is an art to calculate and accurately measure things, the existence of which cannot be proved. Even Guygens recognized that he did not understand the meaning of the decent differences.

As the irony of fate can be considered an appearance in the middle of a century a non-standard analysis, which proved that the initial point of view is actual infinitely small - also consistent and could be based on the basis of the analysis.

Definition of infinitely small and infinitely big feature

Let X. 0 There is a finite or infinitely remote point: ∞, -∞ or + ∞.

Definition of infinitely small functions
Function α. (x) called infinitely small with x seeking x 0 0 and it is equal to zero:
.

Definition of infinitely big function
Function F. (x) called infinitely big with x seeking x 0 if the function has a limit at x → x 0 and it is equal to infinity:
.

Properties of infinitely small functions

Property of the amount, difference and works of infinitely small functions

Amount, difference and work finite number of infinitely small functions at x → x 0 is an infinitely small function at x → x 0 .

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

Theorem on the product of limited function on infinitely small

Product limited On some punctured neighborhood point x 0 , infinitely small, with x → x 0 is an infinitely small function at x → x 0 .

Property for the presentation of a function in the form of a permanent and infinitely small function

In order to function f (x) had the end limit, it is necessary and enough to
,
where - infinitely small function at x → x 0 .

Properties of infinitely large functions

The theorem on the sum of limited function and infinitely large

Amount or difference limited function, on some punctured point x 0 , and infinitely a big function, with x → x 0 is an infinitely large function at x → x 0 .

The theorem on the private from dividing a limited function to infinitely large

If F. (x) is infinitely big at x → x 0 , and function g (x) - limited to some punctured point x 0 T.
.

The theorem on the private from division limited from the bottom of the function is infinitely small

If the function, on some punctured point neighborhood, in absolute value is limited to the bottom with a positive number:
,
And the function is infinitely small at x → x 0 :
,
and there is a pouring neighborhood of a point on which, then
.

Property of inequalities of infinitely large functions

If the function is infinitely large with:
,
and functions and, on some punctured neighborhood, satisfy the inequality:
,
The function is also infinitely large at:
.

This property has two special cases.

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

The relationship between infinitely large and infinitely small functions

Of the two previous properties, the relationship flows between infinitely large and infinitely low functions.

If the function is infinitely large, then the function is infinitely small at.

If the function is infinitely small with, and the function is infinitely large at.

The relationship between an infinitely small and infinitely high function can be expressed symbolic:
, .

If the infinitely small function has a certain sign, that is, positive (or negative) on some punctured point neighborhood, then you can write like this:
.
Similarly, if an infinitely large function has a certain sign, then they write:
, or .

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

Additional formulas connecting infinity symbols can be found on the page.
"Infinitely remote points and their properties."

Proof of properties and theorems

Proof of the theorem on the product of limited function on infinitely small

To prove this theorem, we will use. And also use the property of infinitely small sequences, according to which

Let the function be infinitely small at, and the function is limited in some punctured point:
at.

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

.
,
A sequence is infinitely small:
.

We use the fact that the product of the limited sequence is infinitely small there is an infinitely small sequence:
.
.

Theorem is proved.

Proof Properties on the presentation of a function in the form of a permanent and infinitely small function

Necessity. Let the function be at the final limit at the point
.
Consider a function:
.
Using the function difference limit property, we have:
.
That is, there is an infinitely small function at.

Adequacy. Let it be. Apply the property limit property:
.

Property is proved.

Proof of the theorem on the sum of limited function and infinitely large

To proof theorem, we will use the definition of the limit of the function by Heine

at.

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

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

Since the amount or difference between limited sequences and infinitely large
.
Then, according to the determination of the heine sequence limit,
.

Theorem is proved.

Proof of the theorem on private from dividing limited function to infinitely large

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

Let the function be infinitely large at, and the function is limited in some punctured point:
at.

Since the function is infinitely large, then there is a joining the neighborhood of the point on which it is defined and does not appeal to zero:
at.
Let there be an intersection of the surroundings and. Then the functions are defined on it.

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

Since the private sequence from dividing the limited sequence is infinitely large is an infinitely small sequence,
.
Then, according to the determination of the heine sequence limit,
.

Theorem is proved.

The proof of the partial theorem from dividing the function limited to the bottom is infinitely small

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

Let the function be infinitely small at, and the function is limited by an absolute value from the bottom with a positive number, on some punctured neighborhood of the point:
at.

By condition, there is a punching neighborhood of a point on which the function is determined and does not appeal to zero:
at.
Let there be an intersection of the surroundings and. Then the functions are defined on it. And and.

Let there be an arbitrary sequence converging to whose elements belong to the neighborhood:
.
Then the sequences are defined and. Moreover, the sequence is limited to the bottom:
,
And the sequence is infinitely small with zero members:
, .

Since private from dividing limited from the bottom of the sequence is infinitely small is an infinitely large sequence,
.
And let it have a piercing neighborhood of a point on which
at.

Take an arbitrary sequence convergent to. Then, starting from some number n, the sequence elements will belong to this neighborhood:
at.
Then
at.

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

Property is proved.

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