Definitions of finite and endless limits of the function on infinity by Cauchy. Definitions of bilateral and unilateral limits (left and right). Examples of tasks solutions in which, using the definition of Cauchy, it is required to show that the limit on infinity is equal to the specified value ,.

See also: Point surroundings
Universal determination of the limit of the function of Heine and Cauchy

The final limit of the function on infinity

Limit function at infinity:
| F (X) - A |< ε при |x| > N.

Cauchy limit definition
The number A is called the limit of the function F. (x) with x seeking infinity () if
1) There is such | x | \u003e
2) for any, arbitrarily small positive number ε > 0 , There is such a number N ε \u003e K.depending on ε that for all x, | x | \u003e N ε, the values \u200b\u200bof the function belong to ε - the neighborhood of the point A:
| F. (x) - A |< ε .
The limit of the function on infinity is indicated as follows:
.
Or at.

Also used the following designation:
.

We write this definition using logical symbols of existence and universality:
.
Here it is understood that the values \u200b\u200bbelong to the function definition area.

One-sided limits

Left limit function at infinity:
| F (X) - A |< ε при x < -N

There are often cases when the function is defined only for positive or negative values \u200b\u200bof the variable x (more precisely in the neighborhood of the point or). Also limits on infinity for positive and negative values \u200b\u200bx can have different values. Then use one-sided limits.

Left limit in an infinitely remote point or the limit with X seemed to minus infinity () is determined like this:
.
Right limit at an infinitely remote point or the limit for x seeking to plus infinity ():
.
One-sided limits on infinity are often indicated as follows:
; .

Infinite limit of function on infinity

The infinite limit of the function at infinity:
| f (x) | \u003e M with | x | \u003e N.

Definition of an infinite Cauchy limit
Function limit F. (x) When X, seeking infinity (), is equal to infinity, if a
1) There is such a neighborhood of an infinitely remote point | x | \u003e K, on \u200b\u200bwhich the function is determined (here k is a positive number);
2) for any, arbitrarily large number M > 0 , there is such a number N M \u003e K.depending on M, what for all x, | x | \u003e N M, the values \u200b\u200bof the function belong to the neighborhood of an infinitely remote point:
| F. (x) | \u003e M..
The infinite limit for x seeming infinity is indicated as follows:
.
Or at.

With the help of logical symbols of existence and universality, the definitive limit of the function can be written as follows:
.

Similarly, the definitions of infinite limits of certain signs equal to and are introduced.
.
.

Definitions of one-sided limits on infinity.
Left limits.
.
.
.
Right limits.
.
.
.

Determining the limit of the function by Heine

The number A (finite or infinitely remote) is called the limit function f (x) At point X. 0 :
,
if a
1) There is such a neighborhood of an infinitely remote point x 0 on which the function is defined (here or or or);
2) for any sequence (x n)converging to x 0 : ,
The elements of which belong to the neighborhood, sequence (F (x n)) converges to A:
.

If you take the surroundings of an infinitely remote point without a sign as a neighborhood: then we obtain the limit of the function when X seek infinity ,. If you take a left-sided or right-hand neighborhood of an infinitely remote point x 0 : or, we obtain the determination of the limit with X, aspiring to minus infinity and plus infinity, respectively.

The determination of the limit on Heine and Cauchi is equivalent.

Examples

Example 1.

Using Cauchy Definition Show
.

We introduce notation:
.
Find the function definition area. Since the numerator and denominator of the fraction are polynomials, the function is defined for all x besides points in which the denominator adds to zero. Find these points. We solve the square equation. ;
.
Root equations:
; .
Since, then.
Therefore, the function is defined at. This we will use in the future.

We repel the definition of the final limit of the function at infinity by Cauchy:
.
We transform the difference:
.
We divide the numerator and denominator on and multiply on -1 :
.

Let be .
Then
;
;
;
.

So, we found that when
.
.
Hence it follows that
at, and.

Since you can always increase, then take. Then for anyone
at.
It means that .

Example 2.

Let be .
Using the limit to the Cauchy limit to show that:
1) ;
2) .

1) Solution with X striving to minus infinity

Since, the function is defined for all x.
We repel the determination of the limit of the function with equal to minus infinity:
.

Let be . Then
;
.

So, we found that when
.
We introduce positive numbers and:
.
It follows that for any positive number M, there is a number, so that
.

It means that .

2) Solution with X striving to plus infinity

We convert the source function. Multiply the nipper and denominator of the fraction on and applies square difference formula:
.
We have:

.
We repel the definition of the right limit of the function when:
.

We introduce the designation :.
We transform the difference:
.
Multiply the numerator and denominator on:
.

Let be
.
Then
;
.

So, we found that when
.
We introduce positive numbers and:
.
Hence it follows that
when and.

Since it is performed for any positive number, then
.

References:
CM. Nikolsky. Course of mathematical analysis. Volume 1. Moscow, 1983.

In this article you will learn how to solve the limits?

The solution of the limits is one of the important sections of mathematical and computing analysis. Many students and students of universities cope with this problem freely when others constantly ask the same question: "How to solve the limits?". Finding the limits the topic is relevant. There are many ways to solve limits. Identical limits can be found according to the Lopital law and without his help. However, at first we should figure out what the limit is?

The limit has three parts

The first is to all the famous Lim icon, the second, this is what is written under it.

For example: x -\u003e 1. This entry will be read so (X is strive for 1).

The third part is the function itself that stands after the LIM sign.

I would like to clarify, the value of the X is striving for 1, this is the value x.in which h. Takes certain values \u200b\u200bthat are close to one or almost the same.

Establish limits, it's easy if you figure it out.

The first rule of solution limits

If the function is provided to us, simply substitute the number in the function. This is the elementary limits that are really found in the examples and very often.

There are limits where x-\u003e? Then infinity is the function where X is infinitely increasing. The value of such a function is (1). To solve this limit, we need to follow our first rule to substitute the value (1) to the function and get the answer.

From the foregoing to learn to solve the most difficult limits you must remember the rules for the solution of elementary limits.

• Rule first: A function is given, substitute the number to the function.
• Rule second: Infinity is given, we substitute (1) to the function.

As soon as you understand this, immediately begin to notice the elementary limits and can solve them.So we learned to solve the light limits. Now get acquainted with the solution of more complex limits.

There are many limits with? One of these options is the limit? /?

Such a function is possible when x-\u003e?, And the limit is expressed as a fraction.

Many are interested in whether to solve such a limit?

The first thing you have to remember is to find in the numels of x on seniority, i.e. In the greatest degree of all x that is in the numerator.

Lim + (x -\u003e?)? ((2x ^ 2-3x-4) / (3x ^ 2 + 1 + x)) ^?

We see that the eldest degree in the numerator is 2

Now, we need to do the most just with the denominator. In the denominator, the older degree is also 2.

Principle: In order to solve this function, we should also divide and divider to divide on x in the oldest degree in the limit. In the event if it was equal to 2. If the degree of the numerator was 4, and the denominator 2, then we would choose 4. Because it is the most eldest degree in the functions given to us. See how quickly we learned to solve the limits of the species? /?

Now consider the solution of the most difficult limits. This is a type of 0/0.

Such limits are very reminding us the solution of the limits of the infinity of infinity. But there is a difference that is important to remember when solving. When X is striving for infinity, it is infinitely increasing, and here it is equal to 0, i.e. finite number.

To solve a similar function, we should, and numerals and denominator decompose on multipliers. To get an elementary discriminant, known to us from grade 6. Calculate the discriminant and substitute the answers to our function. Find a finite answer.

Rule: If in a numerator or denominator, you can take a kind of number for this bracket, then we, without thinking, be sure to endure.

There are many different ways to solve more complex limits. One of them is the replacement method. Replace any variable easier than constantly lay out on multipliers. Very often this method is used to make the first wonderful limit from the complex limit.

Let's look more detailed by the example.

Example: lim + (x-\u003e 0)? (arctg4x / 7x) ^?

Decision: We see that our function is presented in the form of uncertainty 0/0, which we have passed

Lim + (x-\u003e 0)? (arctg4x / 7x) ^? \u003d 0/0.

We see in the limit Arctangent, a bad function from which we need to get rid of. It will be very comfortable for us if we are Arctangent to turn into one simple and lightweight letter.

We will replace: arctg replace on y. And in the process of solving, Arctangens will be called as y. If our X is striving for zero, Arctangenes we replaced on y, then write down that y also strives for zero. All that we have remained in the denominator to express X through the cheeky. To do this, in both parts of equality we add TG

Expressions will acquire this species:

TG (arctg4x) \u003d TGY

On the left side, two functions we remove, they are converged and disappear.

We still have:

4x \u003d TGU, hence: x \u003d TGY / 4

And now the most elementary:

Lim + (x-\u003e 0)? (Y / (7 * TGY / 4)) ^?

Go ahead. Within there is not only one wonderful limit, and they are two. Now we will not only figure it out with the concept of a second wonderful limit, but also learn to decide it. Is the second wonderful limit exist to solve the uncertainty of the species 1 ^? In mathematics, it is written so a (x) -\u003e? This type of this function is the easiest, there are functions and more difficult, the most important thing that she sighs to infinity.

It should be remembered that as soon as our limit is to a degree, it is the main sign that such an expression will help us solve the second wonderful limit. Now we will discuss in more detail on the example, which is found very often, I advise him to learn in detail.

Dan us limit: Lim + (x -\u003e?)? ((x-2) / (x + 1)) ^ (2x + 3)?

This type of type (?/?)^ ? The second wonderful limit does such a species decides, as we know, he solves the type of 1 ^?, For this, our function must be transformed into another look. In the denominator, we see X + 1, it means that there should also be x + 1 in the numerator

Lim + (x -\u003e?)? ((x + 1-3) / (x + 1)) ^ (2x + 3)?

Now to us it is necessary to smooth the numerator to the denominator. Then our foundation will be similar to our uncertainty, but there is a minus sign that bothers us. We do fraction with three floors and see our uncertainty? /?. And we already know how to calculate such a function. We divide both parts of the fraction on x, and ready. We have a response.

I want to congratulate you, dear readers, you learned to solve the limits. I hope my article was informative, exciting and interesting!

Elementary functions and their graphics.

The main elementary functions are: a power function, an indicative function, a logarithmic function, trigonometric functions and inverse trigonometric functions, as well as a polynomial and rational function, which is the ratio of two polynomials.

Elementary functions include those functions that are obtained from elementary by applying the main four arithmetic action and the formation of a complex function.

Charts of elementary functions

	Straight line - schedule of linear function y \u003d AX + B. The function y monotonically increases at a\u003e 0 and decreases at a< 0. При b = 0 прямая линия проходит через начало координат т. 0 (y = ax - прямая пропорциональность)
	Parabola - Schedule the function of square triple y \u003d ah 2 + bx + with. It has a vertical axis of symmetry. If a\u003e 0, has a minimum, if< 0 - максимум. Точки пересечения (если они есть) с осью абсцисс - корни соответствующего квадратного уравнения aX 2 + BX + C \u003d 0
	Hyperbola - Function schedule. At a\u003e o is located in the I and III of the quarters, with< 0 - во II и IV. Асимптоты - оси координат. Ось симметрии - прямая у = х(а > 0) or y - - x (a< 0).
	Exponential function. Exhibitor (indicative function on the basis of E) y \u003d e x. (Other Writing y \u003d Ехр (x)). Asymptotta - the abscissa axis.
	Logarithmic function y \u003d log a x (A\u003e 0)
	y \u003d sinx. Sinusoid - Periodic function with a period T \u003d 2π

Limit function.

The function y \u003d f (x) has the number A of the limit when the X K A, if for any number ε\u003e 0 there is such a number δ\u003e 0, which | Y - A | \u003cΕ if | x - a | \u003cΔ,

or lim y \u003d a

Continuity function.

The function y \u003d f (x) is continuous at point x \u003d A, if Lim f (x) \u003d f (a), i.e.

the limit of the function at the point X \u003d A is equal to the value of the function at this point.

Finding the limits of functions.

The main theorems about the limits of functions.

1. The limit of the constant value is equal to this constant:

2. The limit of the algebraic amount is equal to the algebraic sum of the limits of these functions:

lim (F + G - H) \u003d Lim F + Lim G - Lim H

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

lim (F * G * H) \u003d Lim F * Lim G * Lim H

4. The limit of the private two functions is equal to the private limits of these functions, if the denominator limit is not equal to 0:

lim ------- \u003d ----------

First wonderful limit: Lim --------- \u003d 1

The second wonderful limit: Lim (1 + 1 / x) x \u003d E (E \u003d 2, 718281 ..)

Examples of finding functions.

5.1. Example:

Any limit consists of three parts:

1) All the known limit badge.

2) records under the limit icon. The record read "X is striving for one." Most often - it is x, although instead of "IKSA" there may be any other variable. At the site of the unit there may be a completely any number, as well as infinity 0 or.

3) functions under the sign of the limit, in this case.

Record itself It is read like this: "The limit of the function at X is seeking to one."

Very important question - what does the expression "X strive to unit "? Expression "ix strive to unity "should be understood as" X "consistently takes values, which are infinitely close to one and almost one of them coincide.

How to solve the above example? Based on the foregoing, it is necessary to simply substitute a unit into a function standing under the sign of the limit:

So, the first rule : When the limit is given, you must first simply substitute the number to the function.

5.2. Example with infinity:

We understand what? This is the case when it is indefinitely increasing.

So: if , then function she strives for minus infinity:

According to our first rule, we substitute instead of "IKSA" to the function infinity and get the answer.

5.3. Another example with infinity:

Again, we begin to increase to infinity, and look at the behavior of the function.
Conclusion: the prifunction is increasing

5.4. Series of examples:

Try to independently analyze the following examples and solve the simplest types of limits:

, , , , , , , , ,

What needs to be remembered and understand from the foregoing?

When any limit is given, you first simply substitute the number into the function. At the same time, you must understand and immediately solve the simplest limits, such as , , etc.

6. Limits with the uncertainty of the type and solving them.

Now we will look at the group of limits when, and the function is a fraction in the numerator and the denominator of which are polynomials.

6.1. Example:

Calculate the limit

According to our rule, facing infinity to the function. What do we get at the top? Infinity. And what happens at the bottom? Also infinity. Thus, we have the so-called uncertainty of the species. It would be possible to think that \u003d 1, and the answer is ready, but in general it is not at all, and you need to apply some decisions that we now consider.

How to solve the limits of this type?

First we look at the numerator and find to the high degree:

The older degree in the numerator is two.

Now we look at the denominator and also find to the high degree:

The older degree of denominator is equal to two.

Then we choose the most eldest degree of the numerator and the denominator: in this example they coincide and equal to twice.

So, the solution method is as follows: in order to reveal uncertainty it is necessary to divide the numerator and denominator on in high degree.

Thus, the answer, and not at all.

Example

Find a limit

Again in the numerator and denominator we find to the top degree:

Maximum degree in numeric: 3

Maximum degree in denominator: 4

Choose most The value in this case is four.
According to our algorithm, to disclose uncertainty divide the numerator and denominator on.

Example

Find a limit

Maximum degree of "IKSA" in the Numerator: 2

Maximum degree "IKSA" in the denominator: 1 (can be written as)
To disclose uncertainty, it is necessary to divide the numerator and denominator on. The finishing solution may look like this:

We divide the numerator and denominator on

Determining the limits of the sequence and function, properties of the limits, the first and second wonderful limits, examples.

Constant number but called limit sequences(x n), if for any arbitrarily small positive number ε\u003e 0 there is N number that all values x N. which n\u003e n satisfy inequality

This is written as follows: or x n → a.

Inequality (6.1) is equivalent to dual inequality

a - ε.< x n < a + ε которое означает, что точки x N.From some number N\u003e N, lie inside the interval (A-ε, A + ε), i.e. Include in which small ε-neighborhood of the point but.

The sequence having a limit is called converging, otherwise - Diverging.

The concept of the limit of the function is a generalization of the concept of the sequence limit, since the sequence limit can be considered as the limit of the function x n \u003d f (n) of the integer argument n..

Let the function f (x) and let a. - limit point Areas of defining this function D (F), i.e. such a point, any neighborhood of which contains the points of the set D (F) other than a.. Point a. It may belong to the set D (F), and may not belong to it.

Definition 1. Constant number and called limit functions f (x) for X → A, if for any sequence (x n) of the values \u200b\u200bof the argument striving for butcorresponding to them sequences (F (x n)) have the same limit A.

This definition is called determination of the limit of the function by Heine or " in the language of sequences”.

Definition 2.. Constant number and called limit functions f (x) for X → A, if, setting an arbitrary, as an empty positive number ε, can be found δ\u003e 0 (depending on ε), which for all x.lying in the ε-neighborhood of the number but. for x.satisfying inequality
0 < x-a < ε , значения функции f(x) будут лежать в ε-окрестности числа А, т.е. |f(x)-A| < ε

This definition is called determining the limit of the function on Cauchy,or "In the language ε - δ"

Definitions 1 and 2 are equivalent. If the function f (x) at x → a has limitequal to a, it is written in the form

In the event that the sequence (F (x n)) increases indefinitely (or decreases) in any way of approximation x. To its limit but, then we say that the function f (x) has endless limit And write it in the form:

Variable value (i.e. sequence or function), the limit of which is zero, is called infinitely low magnitude.

The variable value whose limit is infinity, is called infinitely large magnitude.

To find the limit in practice enjoy the following theorems.

Theorem 1. . If every limit exists

(6.4)

(6.5)

(6.6)

Comment. Expressions of the form 0/0, ∞ / ∞, ∞-∞ 0 * ∞ are uncertain, for example, the ratio of two infinitely small or infinitely large values, and finding the limit of this type is called "Disclosure of uncertainty".

Theorem 2.

those. You can move to the limit at the bottom of the degree at a constant indicator, in particular,

Theorem 3.

(6.11)

where E.» 2.7 - The basis of the natural logarithm. Formulas (6.10) and (6.11) are called the first wonderful and the second wonderful limit.

Used in practice and consequences of formula (6.11):

(6.12)

(6.13)

(6.14)

in particular the limit

If x → a and at the same time x\u003e a, then x → a + 0 write. If, in particular, a \u003d 0, then instead of the character 0 + 0 write +0. Similarly, if x → a and at the same time x and are called respectively the limit on the right and limit of the left functions f (x) At point but. To exist the limit function f (x) at X → A, it is necessary and enough to . Function F (x) is called Continuous at pointx 0 if the limit

(6.15)

Condition (6.15) can be rewritten in the form:

that is, the limit is possible under the sign of the function, if it is continuous at this point.

If equality (6.15) is broken, then they say that for x \u003d x o function f (x) it has gap.Consider the function y \u003d 1 / x. The area of \u200b\u200bdefinition of this function is the set R., except x \u003d 0. Point x \u003d 0 is the limit point of the set D (F), since in any neighborhood, i.e. In any open interval containing a point 0, there are points from D (F), but it does not belong to this set. The value f (x O) \u003d f (0) is not defined, therefore at point x o \u003d 0 the function has a break.

Function F (x) is called continuous on the right at the point x o if the limit

and continuous left at point x o if the limit

Continuity function at point x O. It is equivalent to its continuity at this point at the same time on the right and left.

In order for the function to be continuous at the point x O.For example, on the right, it is necessary, firstly, to exist a final limit, and secondly, that this limit is equal to f (x o). Therefore, if at least one of these two conditions is performed, the function will have a gap.

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

2. If the limit is + ∞ or -∞ or does not exist, they say that in point X O. The function has a break second race.

For example, the function y \u003d CTG x at x → +0 has a limit equal to + ∞, it means that at point x \u003d 0 it has a gap of the second kind. Function y \u003d e (x) (whole part of x.) At points with whole abscissions, the first kind of gaps, or jumps.

Function, continuous at each point of the gap, is called Continuous in . Continuous function is depicted with a solid curve.

To the second remarkable limit, many tasks related to the continuous growth of any value are given. Such tasks, for example, include: the growth of the contribution under the law of complex interest, the growth of the population of the country, the disintegration of the radioactive substance, the reproduction of bacteria, etc.

Consider example Ya. I. Perelmangiving an interpretation of the number e. In the task of complex percentages. Number e.there is a limit . In Sberbanks, interest money joins the principal capital annually. If the accession is performed more often, the capital grows faster, as a large amount is involved in the education of interest. Take a purely theoretical, very simplified example. Let 100 den go to the bank. units. At the rate of 100% per annum. If interest money will be attached to the main capital only after the expiration of the year, then to this term 100 den. units. turns into 200 den. Let's see now, what will turn 100 den. un., if interest money to attach every six months to the main capital. After half of the year 100 den. units. It will grow in 100 × 1.5 \u003d 150, and after half a year - in 150 × 1,5 \u003d 225 (den. units). If the join to do every 1/3 year, then after the year of 100 den. units. turns into 100 × (1 +1/3) 3 ≈ 237 (den. units). We will participate the time to attach interest money to 0.1 years, to 0.01 years, up to 0.001 years, etc. Then out of 100 den. units. A year later, it will turn out:

100 × (1 +1/10) 10 ≈ 259 (den. Units),

100 × (1 + 1/100) 100 ≈ 270 (den. Units),

100 × (1 + 1/1000) 1000 ≈271 (den. Units).

In case of limitless reduction in the time of interest, the increasing capital is not increasingly increasing, but approximately 271 is approximately 2.71 times. Capital, laid under 100% per annum, cannot increase second because the limit

Example 3.1.. Using the determination of the limit of the numerical sequence, to prove that the sequence x n \u003d (n - 1) / n has a limit equal to 1.

Decision. We need to prove that, whatever ε\u003e 0, we will have a natural number n, such that for all N\u003e N is inequality | x n -1 |< ε

Take any ε\u003e 0. Since x n -1 \u003d (n + 1) / n - 1 \u003d 1 / n, then for finding N it suffices to solve the inequality 1 / n<ε. Отсюда n>1 / ε and, therefore, for n, it can be taken as a whole of 1 / ε n \u003d e (1 / ε). We thus proved that the limit.

Decision. Apply the theorem of the amount of the amount and find the limit of each terms. For n → ∞, the numerator and denominator of each term tends to infinity, and we cannot directly apply the theorem of the Private Limit. Therefore, we first convert x N., dividing the numerator and denominator of the first term on n 2.and second on n.. Then, applying the theorem the limit of the private and limit of the amount, we will find:

Example 3.3.. . To find .

Here we took advantage of the degree limit theorem: the limit of the degree is equal to the foundation limit.

Example 3.4.. To find ( ).

Decision. Apply the theorem the difference limit is impossible, since we have uncertainty of the form ∞-∞. We transform a general member formula:

Example 3.5.. The function f (x) \u003d 2 1 / x is given. Prove that the limit does not exist.

Decision. We use the definition of the 1 limit of the function through the sequence. Take the sequence (x n) convergent to 0, i.e. We show that the value f (x n) \u003d for different sequences behaves differently. Let x n \u003d 1 / n be. Obviously, then the limit Choose now as x N. The sequence with a general member x n \u003d -1 / n, also striving for zero. Therefore, the limit does not exist.

Example 3.6.. Prove that the limit does not exist.

Decision. Let x 1, x 2, ..., x n, ... - the sequence for which
. How the sequence behaves (f (x n)) \u003d (sin x n) at different x n → ∞

If x n \u003d p n, then sin x n \u003d sin (p n) \u003d 0 for all n. and limit if
x n \u003d 2p n + p / 2, then sin x n \u003d sin (2 p n + p / 2) \u003d sin p / 2 \u003d 1 for all n. And therefore the limit. Thus, it does not exist.

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