is given, in other words, known, if for each value of the possible number of arguments it is possible to find out the corresponding value of the function. The most common three function definition method: tabular, graphic, analytical, there are also verbal and recursive methods.

1. Tabular way the most widespread (tables of logarithms, square roots), its main advantage is the possibility of obtaining a numerical value of the function, the disadvantages are that the table can be difficult to read and sometimes does not contain intermediate values ​​of the argument.

For example:

x
y

Argument X takes the values ​​specified in the table, and at defined according to this argument X.

2. Graphical way consists in drawing a line (graph), in which the abscissas represent the values ​​of the argument, and the ordinates represent the corresponding values ​​of the function. Often, for clarity, the scales on the axes are taken different.

For example: to find the schedule at, which corresponds to x = 2.5 it is necessary to draw a perpendicular to the axis X at the mark 2,5 . The mark can be quite accurately done with a ruler. Then we find that at X = 2,5 at equals 7,5 , but if we need to find the value at at X equal to 2,76 , then the graphical way of setting the function will not be accurate enough, because The ruler does not allow for such an accurate measurement.

The advantages of this method of setting functions are in the ease and integrity of perception, in the continuity of the change of the argument; the disadvantage is a decrease in the degree of accuracy and the difficulty of obtaining accurate values.

3. Analytical method consists in specifying a function by one or more formulas. The main advantage of this method is the high accuracy of determining the function of the argument of interest, and the disadvantage is the time spent on additional mathematical operations.

For example:

The function can be specified using the mathematical formula y=x2, then if X equals 2 , then at equals 4, we are building X into a square.

4. verbal way consists in defining the function in plain language, i.e. words. In this case, it is necessary to give input, output values ​​and the correspondence between them.

For example:

You can verbally specify a function (task) that is accepted as a natural argument X with the corresponding value of the sum of the digits that make up the value at. Explain: if X equals 4 , then at equals 4 , what if X equals 358 , then at is equal to the sum 3 + 5 + 8 , i.e. 16 . Further similarly.

5. Recursive way consists in specifying a function through itself, while function values are defined in terms of its other values. This way of defining a function is used in defining sets and series.

For example:

When decomposed Euler numbers given by the function:

Its abbreviation is given below:

In direct calculation, infinite recursion occurs, but it can be proved that the value f(n) with increasing n tends to unity (therefore, despite the infinity of the series , the value Euler numbers certainly). For an approximate calculation of the value e it is enough to artificially limit the recursion depth to some predetermined number and, upon reaching it, use it instead f(n) unit.

What do the words mean "set function"? They mean: to explain to everyone, about what specific function is talking. Moreover, explain clearly and unambiguously!

How can I do that? How set a function?

You can write a formula. You can draw a graph. You can make a table. Any way is some rule by which you can find out the value of the player for the x value we have chosen. Those. "set function", this means - to show the law, the rule according to which x turns into a y.

Usually, in a variety of tasks there are ready functions. They give us already set. Decide for yourself, but decide.) But ... Most often, schoolchildren (and students) work with formulas. They get used to it, you understand... They get used to it so much that any elementary question related to a different way of specifying a function immediately upsets a person...)

To avoid such cases, it makes sense to understand the different ways of defining functions. And, of course, apply this knowledge to "tricky" questions. It's simple enough. If you know what a function is...)

Go?)

Analytical way of defining a function.

The most versatile and powerful way. Function defined analytically, this is the function that is given formulas. Actually, this is the whole explanation.) Familiar to everyone (I want to believe!)) functions, for example: y=2x or y=x2 etc. etc. are given analytically.

By the way, not every formula can define a function. Not every formula follows the strict condition of the function definition. Namely - for each x there can only be one game. For example, in the formula y = ±x, for one values ​​x=2, it turns out two y values: +2 and -2. It is impossible to define a single-valued function with this formula. And with multivalued functions in this section of mathematics, in mathematical analysis, they do not work, as a rule.

Why is the analytical way of defining a function good? The fact that if you have a formula - you know about the function all! You can make a table. Build a graph. Explore this feature in full. Predict exactly where and how this function will behave. All mathematical analysis rests on this method of defining functions. Let's say it's extremely difficult to take the derivative of a table...)

The analytical method is quite familiar and does not create problems. Except perhaps some varieties of this method that students encounter. I'm talking about parametric and implicit assignment of functions.) But such functions are in a special lesson.

Let's move on to less familiar ways of defining a function.

Tabular way of defining a function.

As the name suggests, this method is a simple plate. In this table, each x corresponds to ( is aligned) some value of the player. The first line contains the values ​​of the argument. The second line contains the corresponding function values, for example:

Table 1.

x	- 3	- 1	0	2	3	4
y	5	2	- 4	- 1	6	5

Please pay attention! In this example, y depends on x anyhow. I came up with this on purpose.) There is no pattern. It's okay, it happens. Means, exactly I set this particular function. Exactly I set up a rule by which x turns into a y.

Can be compiled another a plate with a pattern. This plate will set another function, for example:

Table 2.

x	- 3	- 1	0	2	3	4
y	- 6	- 2	0	4	6	8

Did you catch the pattern? Here, all values ​​of the y are obtained by multiplying x by two. Here is the first "tricky" question: can the function specified using Table 2 be considered a function y = 2x? Think for a while, the answer will be below, in a graphical way. It's very clear there.)

What is good tabular way of setting a function? Yes, you don't have to count anything. Everything has already been calculated and written in the table.) And there is nothing more good. We do not know the value of the function for x, which are not in the table. In this method, such x values ​​\u200b\u200bare simply does not exist. By the way, this is a clue to the tricky question.) We can't find out how the function behaves outside the table. We can't do anything. Yes, and the visibility in this method leaves much to be desired ... For clarity, a graphical method is good.

Graphical way to define a function.

In this method, the function is represented by a graph. The argument (x) is plotted along the abscissa, and the value of the function (y) is plotted along the ordinate. According to the schedule, you can also choose any X and find the corresponding value at. The schedule can be any, but... not any.) We work only with single-valued functions. The definition of such a function clearly states: each X is aligned the only one at. One one, not two, or three... For example, let's look at the circle graph:

A circle is like a circle... Why shouldn't it be a graph of a function? And let's find which y will correspond to the value of x, for example, 6? We move the cursor over the chart (or touch the picture on the tablet), and ... we see that this X corresponds to two player values: y=2 and y=6.

Two and six! Therefore, such a graph will not be a graphic assignment of a function. On the one x accounted for two game. This graph does not correspond to the definition of the function.

But if the uniqueness condition is met, the graph can be absolutely anything. For example:

This very krivulina - and there is a law by which you can translate x into a y. Unambiguous. We would like to know the value of the function for x = 4, for example. We need to find the four on the x-axis and see which y corresponds to this x. Hover the mouse over the figure and see that the value of the function at for x=4 equals five. We do not know by what formula such a transformation of X into Y is given. And it is not necessary. Everything is set by the schedule.

Now we can return to the "tricky" question about y=2x. Let's plot this function. Here he is:

Of course, when drawing this graph, we did not take an infinite number of values X. We took several values, counted y, made a plate - and you're done! The most literate generally took only two values ​​​​of X! And rightly so. For a straight line, you don't need more. Why extra work?

But we knew exactly what x can be anyone. Whole, fractional, negative... Any. This is according to the formula y=2x it is seen. Therefore, we boldly connected the points on the graph with a solid line.

If the function is given to us by Table 2, then we will have to take x-values only from the table. For other Xs (and Ys) are not given to us, and there is nowhere to take them. There are none, these values, in this function. Schedule will turn out from points. We point the mouse at the picture and see the graph of the function given by Table 2. I didn’t write the x-y values ​​​​on the axes, will you figure it out, go, by the cells?)

Here is the answer to the tricky question. Function given by Table 2 and function y=2x - various.

The graphical method is good for its clarity. You can immediately see how the function behaves where it increases. where it decreases. From the graph, you can immediately find out some important characteristics of the function. And in the topic with the derivative, tasks with graphs - all the time!

In general, analytical and graphical ways of defining a function go hand in hand. Working with the formula helps to build a graph. And the graph often suggests solutions that you won’t notice in the formula ... We will be friends with the graphs.)

Almost any student knows the three ways to define a function that we have just covered. But to the question: "And the fourth one!?" - freezes thoroughly.)

There is such a way.

Verbal description of the function.

Yes Yes! A function can be quite unambiguously defined in words. The great and mighty Russian language is capable of much!) For example, the function y=2x can be given the following verbal description: each real value of the argument x is assigned its doubled value. Like this! The rule is set, the function is set.

Moreover, it is possible to specify a function verbally, which is extremely difficult, if not impossible, to specify by a formula. For example: each value of the natural argument x is assigned the sum of the digits that make up the value of x. For example, if x=3, then y=3. If a x=257, then y=2+5+7=14. And so on. It is difficult to write this down in a formula. But the table is easy to make. And build a chart. By the way, the schedule turns out to be funny ...) Try it.

The method of verbal description is a rather exotic method. But sometimes it happens. Here I brought it to give you confidence in unexpected and non-standard situations. You just need to understand the meaning of the words "function set..." Here is the meaning:

If there is a law of one-to-one correspondence between X and at means there is a function. What law, in what form it is expressed - by a formula, a tablet, a graph, words, songs, dances - does not change the essence of the matter. This law allows you to determine the corresponding value of y by the value of x. Everything.

Now we will apply this deep knowledge to some non-standard tasks.) As promised at the beginning of the lesson.

Exercise 1:

The function y = f(x) is given in Table 1:

Table 1.

Find the value of the function p(4) if p(x)= f(x) - g(x)

If you can't figure out what's what at all - read the previous lesson "What is a function?" There, it is very clearly written about such letters and brackets.) And if only the tabular form confuses you, then we’ll figure it out here.

It is clear from the previous lesson that if, p(x) = f(x) - g(x), then p(4) = f(4) - g(4). Letters f and g mean the rules according to which each X is assigned its own Y. For each letter ( f and g) - own rule. Which is given by the corresponding table.

Function value f(4) determined from Table 1. This will be 5. The value of the function g(4) determined by Table 2. This will be 8. The most difficult remains.)

p(4) = 5 - 8 = -3

This is the correct answer.

Solve the inequality f(x) > 2

That's it! It is necessary to solve the inequality, which (in the usual form) is brilliantly absent! It remains to either quit the task, or turn on the head. We choose the second and argue.)

What does it mean to solve an inequality? This means to find all the values ​​of x for which the condition given to us is satisfied f(x) > 2. Those. all function values ​​( at) must be greater than two. And we have every y on the chart... And there are more than two, and less... And let's, for clarity, draw a line on this two! We move the cursor over the picture and see this border.

Strictly speaking, this boundary is the graph of the function y=2, but that's not the point. It is important that now on the graph it is very clearly visible where, at what x, function values, i.e. y, more than two. They are more X > 3. At X > 3 our entire function passes above borders y=2. That's the whole solution. But it's still too early to turn off your head!) We still need to write down the answer ...

The graph shows that our function does not extend left and right to infinity. The points at the ends of the graph speak about this. The function ends there. Therefore, in our inequality, all x's that go beyond the limits of the function have no meaning. For the function of these x's does not exist. And we, in fact, solve the inequality for the function ...

The correct answer would be:

3 < X ≤ 6

Or, in another form:

X ∈ (3; 6]

Now everything is as it should be. The triple is not included in the answer, because the original inequality is strict. And the six turns on, because and the function at six exists, and the inequality condition is satisfied. We have successfully solved an inequality that (in its usual form) does not exist...

This is how some knowledge and elementary logic save in non-standard cases.)

Functions can be defined in a variety of ways. However, the following three ways of defining functions are most common: analytical, tabular, and graphical.

Analytical way of defining a function. With the analytical method of setting, the function is defined using an analytical expression, that is, using a formula that indicates what operations must be performed on the value of the argument in order to obtain the corresponding value of the function.

In Sections 2 and 3, we have already met with functions defined with the help of formulas, i.e., analytically. At the same time, in paragraph 2 for the function, the domain of definition ) was established based on geometric considerations, and for the function, the domain of assignment was indicated in the condition. In Section 3, for the function, the domain of definition was also specified by condition. However, very often a function is specified only with the help of an analytical expression (formula), without any additional conditions. In such cases, by the domain of a function, we mean the set of all those values ​​of the argument for which this expression makes sense and leads to the actual values ​​of the function.

Example 1. Find the scope of a function

Solution. The function is defined only by a formula, its scope is not specified, and there are no additional conditions. Therefore, under the domain of this function, we must understand the totality of all those values ​​of the argument for which the expression has real values. For this there should be . Solving this inequality, we come to the conclusion that the domain of this function is the segment [-1.1].

Example 2. Find the scope of a function.

Solution. The domain of definition, obviously, consists of two infinite intervals, since the expression does not and makes sense when a is defined for all other values.

The reader will now easily see for himself that for a function, the domain of definition will be the entire numerical axis, and for a function, an infinite interval

It should be noted that it is impossible to identify a function and a formula with which this function is specified. Using the same formula, you can define different functions. Indeed, in Section 2 we considered a function with a domain of definition, in Section 3 a graph was constructed for a function with a domain of definition . And, finally, we have just considered a function defined only by a formula without any additional conditions. The scope of this function is the entire number axis. These three functions are different because they have different scopes. But they are set using the same formula.

The reverse case is also possible, when one function in different parts of its domain of definition is given by different formulas. For example, consider a function y defined for all non-negative values ​​as follows: for at i.e.

This function is defined by two analytic expressions acting on different parts of its domain of definition. The graph of this function is shown in Fig. eighteen.

Tabular way of defining a function. When a function is specified in a table, a table is created in which a number of argument values ​​​​and corresponding function values ​​\u200b\u200bare indicated. Logarithmic tables, tables of values ​​of trigonometric functions and many others are widely known. Quite often it is necessary to use tables of function values ​​obtained directly from experience. The following table shows the resistivities of copper obtained from experience (in cm - centimeters) at various temperatures t (in degrees):

Graphical way to define a function. When a graphical task is given, the graph of the function is given, and its values ​​corresponding to certain values ​​of the argument are directly found from this graph. In many cases, such graphs are drawn using self-recording instruments.

The main ways of specifying functions are given: explicit analytical; interval; parametric; implicit; defining a function using a series; tabular; graphic. Examples of the application of these methods

There are the following ways to define the function y = f (x):

1. An explicit analytical method using a formula of the form y = f (x).
2. Interval.
3. Parametric: x = x (t) , y = y(t).
4. Implicit, as a solution to equation F (x, y) = 0.
5. In the form of a series composed of known functions.
6. Tabular.
7. Graphic.

Explicit way to define a function

At explicit way, the value of the function is determined by the formula, which is the equation y = f (x). On the left side of this equation is the dependent variable y, and on the right side is an expression composed of the independent variable x, constant, known functions, and operations of addition, subtraction, multiplication, and division. Known functions are elementary functions and special functions, the values ​​of which can be calculated using computer technology.

Here are some examples of explicitly defining a function with an independent variable x and a dependent variable y :
;
;
.

Interval way to define a function

At interval method of setting a function, the domain of definition is divided into several intervals, and the function is specified separately for each interval.

Here are some examples of the interval way of defining a function:

Parametric way of defining a function

At parametric method, a new variable is introduced, which is called a parameter. Next, the x and y values ​​are set as functions of the parameter, using the explicit way of setting:
(1)

Here are examples of a parametric way of defining a function using the t parameter:

The advantage of the parametric method is that the same function can be defined in an infinite number of ways. For example, a function can be defined like this:

And it's possible like this:

Such freedom of choice, in some cases, allows you to apply this method to solve equations (see "Differential equations that do not contain one of the variables"). The essence of the application is that we substitute two functions and instead of the variables x and y into the equation. Then we set one of them at our own discretion, so that the other can be determined from the resulting equation.

Also, this method is used to simplify calculations. For example, the dependence of the coordinates of the points of an ellipse with semiaxes a and b can be represented as follows:
.
In a parametric form, this dependence can be given a simpler form:
.

Equations (1) are not the only way to parametrically define a function. You can enter not one, but several parameters by linking them with additional equations. For example, you can enter two parameters and . Then the function definition will look like this:

Here comes an additional equation relating the parameters. If the number of parameters is n , then there must be n - 1 additional equations.

An example of using multiple parameters is set out on the Jacobi Differential Equation page. There, the solution is sought in the following form:
(2) .
The result is a system of equations. To solve it, a fourth parameter t is introduced. After solving the system, three equations are obtained that relate four parameters and .

Implicit way to define a function

At implicit way, the value of the function is determined from the solution of the equation .

For example, the equation for an ellipse is:
(3) .
This is a simple equation. If we consider only the upper part of the ellipse, , then we can express the variable y as a function of x in an explicit way:
(4) .
But even if it is possible to reduce (3) to an explicit way of specifying the function (4), the last formula is not always convenient to use. For example, to find the derivative , it is convenient to differentiate equation (3) rather than (4):
;
.

Setting a function nearby

An extremely important way to define a function is to row representation composed of known functions. This method allows you to explore the function by mathematical methods and calculate its values ​​for applied problems.

The most common representation is to define a function using a power series. In this case, a number of power functions are used:
.
A series with negative exponents is also used:
.
For example, the sine function has the following expansion:
(5) .
Such expansions are widely used in computing to calculate the values ​​of functions, since they allow one to reduce calculations to arithmetic operations.

As an illustration, let's calculate the value of the sine of 30° using expansion (5).
Convert degrees to radians:
.
Substitute in (5):



.

In mathematics, along with power series, expansions into trigonometric series in functions and , as well as in other special functions, are widely used. With the help of series, one can make approximate calculations of integrals, equations (differential, integral, in partial derivatives) and investigate their solutions.

Tabular way of defining a function

At tabular way of setting a function we have a table that contains the values ​​of the independent variable x and the corresponding values ​​of the dependent variable y . The independent and dependent variables may have different designations, but we use x and y here. To determine the value of a function for a given value of x , we use the table to find the value of x that is closest to our value. After that, we determine the corresponding value of the dependent variable y .

For a more precise definition of the value of the function, we consider that the function between two adjacent values ​​of x is linear, that is, it has the following form:
.
Here are the values ​​of the function found from the table, with the corresponding values ​​of the arguments .
Consider an example. Let us need to find the value of the function at . From the table we find:
.
Then

.
Exact value:
.
From this example, it can be seen that the use of linear approximation led to an increase in accuracy in determining the value of the function.

The tabular method is used in applied sciences. Before the development of computer technology, it was widely used in engineering and other calculations. Now the tabular method is used in statistics and experimental sciences to collect and analyze experimental data.

Graphical way to define a function

At graphical way, the value of the function is determined from the graph, along the abscissa axis of which the values ​​of the independent variable are plotted, and along the ordinate axis - the dependent variable.

The graphical method gives a visual representation of the behavior of the function. The results of the study of a function are often illustrated by its graph. From the graph, you can determine the approximate value of the function. This allows you to use the graphical method in applied and engineering sciences.

The concept of a function Ways of defining a function Examples of functions Analytical definition of a function Graphical way of defining a function Limit of a function at a point Tabular way of defining a function Limit theorems Uniqueness of a limit Boundedness of a function that has a limit Passing to a limit at inequality Limit of a function at infinity Infinitesimal functions Properties of infinitesimal functions

The concept of a function is basic and original, as is the concept of a set. Let X be some set of real numbers x. If a certain real number y is assigned to each x ∈ X according to some law, then they say that a function is given on the set X and write. The function introduced in this way is called a numerical one. In this case, the set X is called the domain of the definition of the function, and the independent variable x is called the argument. To indicate a function, sometimes only the symbol is used, which denotes the law of correspondence, i.e. instead of f (x) n and jester, just /. Thus, the function is given if 1) the domain of definition is specified 2) the rule /, which assigns to each value a: € X a certain number y \u003d / (x) - the value of the function corresponding to this value of the argument x. The functions / and g are called equal if their domains of definition coincide and the equality f(x) = g(x) is true for any value of the argument x from their common domain. Thus, the functions y are not equal; they are equal only on the interval [O, I]. Function examples. 1. The sequence (o„) is a function of an integer argument, defined on the set of natural numbers, such that f(n) = an (n = 1,2,...). 2. Function y = n? (read "en-factorial"). Given on the set of natural numbers: each natural number n is associated with the product of all natural numbers from 1 to n inclusive: moreover, 0! = 1. The designation sign comes from the Latin word signum - a sign. This function is defined on the entire number line; the set of its values ​​consists of three numbers -1.0, I (Fig. 1). y = |x), where (x) denotes the integer part of a real number x, i.e. [x| - the largest integer not exceeding It is read: - the game is equal to antie x ”(fr. entier). This function is set on the entire number axis, and the set of all its values ​​consists of integers (Fig. 2). Methods for Specifying a Function Analytical Specifying a Function A function y = f(x) is said to be specified analytically if it is defined using a formula that specifies what operations must be performed on each value of x in order to obtain the corresponding value of y. For example, the function is given analytically. In this case, the domain of the function (if it is not specified in advance) is understood as the set of all real values ​​of the argument x, for which the analytical expression that defines the function takes only real and final values. In this sense, the domain of a function is also called its domain of existence. For the function, the domain of definition is the segment. For the function y - sin x, the domain of definition is the entire numerical axis. Note that not every formula defines a function. For example, the formula does not define any function, since there is not a single real value of x for which both roots written above would have real values. Analytical assignment of a function can look rather complicated. In particular, a function can be defined by different formulas on different parts of its domain of definition. For example, a function could be defined like this: 1.2. Graphical way of specifying a function The function y = f(x) is called specified graphically if its schedule is specified, i.e. a set of points (xy/(x)) on the xOy plane, the abscissas of which belong to the domain of definition of the function, and the ordinates are equal to the corresponding values ​​of the function (Fig. 4). Not for every function, its graph can be depicted in the figure. For example, the Dirichlet function if x is rational, if x is irrational, ZX \o, does not allow such a representation. The function R(x) is given on the entire numerical axis, and the set of its values ​​consists of two numbers 0 and 1. 1.3. Tabular way of specifying a function A function is said to be specified tabular if a table is provided that contains the numerical values ​​of the function for some values ​​of the argument. When a function is defined in a table, its domain of definition consists only of the values ​​x\t x2i..., xn listed in the table. §2. Limit of a function at a point The concept of the limit of a function is central to mathematical analysis. Let the function f(x) be defined in some neighborhood Q of the point xq, except, perhaps, for the extension (Cauchy) point itself. The number A is called the limit of the function f(x) at the point x0 if for any number e > 0, which can be arbitrarily small, there exists a number<5 > 0, such that for all iGH.i^ x0 satisfying the condition the inequality is true Definition of a function Ways of defining a function Examples of functions Analytical definition of a function Graphical way of defining a function Limit of a function at a point Tabular way of defining a function Limit theorems Uniqueness of a limit Boundedness of a function that has a limit Transition to the limit in the inequality Limit of a function at infinity Infinitesimal functions Properties of infinitesimal functions Notation: With the help of logical symbols, this definition is expressed as follows. Examples. 1. Using the definition of the limit of a function at a point, show that the Function is defined everywhere, including the point zo = 1: /(1) = 5. Take any. In order for the inequality |(2x + 3) - 5| took place, it is necessary to fulfill the following inequalities Therefore, if we take we will have. This means that the number 5 is the limit of the function: at point 2. Using the definition of the limit of a function, show that the function is not defined at the point xo = 2. Consider /(x) in some neighborhood of the point-Xq = 2, for example, on the interval ( 1, 5) that does not contain the point x = 0, at which the function /(x) is also not defined. Take an arbitrary number c > 0 and transform the expression |/(x) - 2| for x f 2 as follows For x b (1, 5) we get the inequality From this it is clear that if we take 6 \u003d c, then for all x € (1.5) subject to the condition the inequality will be true This means that the number A - 2 is the limit of a given function at a point Let us give a geometric explanation of the concept of the limit of a function at a point, referring to its graph (Fig. 5). For x, the values ​​of the function /(x) are determined by the ordinates of the points of the curve M \ M, for x > ho - by the ordinates of the points of the curve MM2. The value /(x0) is determined by the ordinate of the point N. The graph of this function is obtained if we take the "good" curve M\MMg and replace the point M(x0, A) on the curve with the point jV. Let us show that at the point x0 the function /(x) has a limit equal to the number A (the ordinate of the point M). Take any (arbitrarily small) number e > 0. Mark on the Oy axis points with ordinates A, A - e, A + e. Denote by P and Q the points of intersection of the graph of the function y \u003d / (x) with the lines y \u003d A - enu = A + e. Let the abscissas of these points be x0 - hx0 + hi, respectively (ht > 0, /12 > 0). It can be seen from the figure that for any x Φ x0 from the interval (x0 - h\, x0 + hi) the value of the function f(x) is between. for all x ⩽ x0 satisfying the condition the inequality is true We set Then the interval will be contained in the interval and, therefore, the inequality or, which will also be satisfied for all x satisfying the condition This proves that Thus, the function y = /(x) has a limit A at the point x0 if, no matter how narrow the e-strip between the lines y = A - eny = A + e, there is such "5 > 0, such that for all x from the punctured neighborhood of the point x0 of the point of the graph of the function y = / (x) are inside the indicated e-band. Remark 1. The quantity b depends on e: 6 = 6(e). Remark 2. In the definition of the limit of a function at the point Xq, the point x0 itself is excluded from consideration. Thus, the value of the function at the Ho ns point does not affect the limit of the function at that point. Moreover, the function may not even be defined at the point Xq. Therefore, two functions that are equal in a neighborhood of the point Xq, excluding, perhaps, the point x0 itself (they may have different values ​​at it, one of them or both together may not be defined), have the same limit for x - Xq, or both have no limit. From this, in particular, it follows that in order to find the limit of a fraction at the point xo, it is legitimate to reduce this fraction by equal expressions that vanish at x = Xq. Example 1. Find The function /(x) = j for all x Ф 0 is equal to one, and at the point x = 0 it is not defined. Replacing f(x) with the function q(x) = 1 equal to it at x 0, we obtain the concept of a function Ways of defining a function Examples of functions Analytical definition of a function Graphical way of defining a function Limit of a function at a point Tabular way of defining a function Limit theorems Uniqueness of a limit Boundedness of a function having a limit transition to the limit in the inequality The limit of a function at infinity Infinitely small functions Properties of infinitely small functions x = 0 limit equal to zero: lim q(x) = 0 (show it!). Therefore, lim /(x) = 0. Problem. Formulate with the help of inequalities (in the language of e -6), which means Let the function /(n) be defined in some neighborhood Π of the point x0, except, perhaps, the point x0 itself. Definition (Heine). The number A is called the limit of the function /(x) at the point x0, if for any sequence (xn) of values ​​of the argument x 6 P, zn / x0) converging to the point x0, the corresponding sequence of values ​​of the function (/(xn)) converges to number A. It is convenient to use the above definition when it is necessary to establish that the function /(x) has no limit at the point x0. To do this, it suffices to find some sequence (/(xn)) that does not have a limit, or to indicate two sequences (/(xn)) and (/(x "n)) that have different limits. Let us show, for example, that the function iiya / (x) = sin j (Fig. 7), defined EVERYWHERE, except for the POINT X = O, Fig. 7 does not have a limit at the point x = 0. Consider two sequences (, converging to the point x = 0. The corresponding sequences values ​​of the function /(x) converge to different limits: the sequence (sinnTr) converges to zero, and the sequence (sin(5 +) converges to one. This means that the function f(x) = sin j at the point x = 0 has no limit. Comment. Both definitions of the limit of a function at a point (Cauchy's definition and Heine's definition) are equivalent. §3. Theorems on limits Theorem 1 (uniqueness of the limit). If the function f(x) has a limit at xo, then this limit is unique. A Let lim f(x) = A. Let us show that no number B φ A can be the limit x-x0 of the function f(x) at the point x0. The fact that lim /(x) φ With the help of logical symbols XO is formulated as follows: Using the inequality we obtain, Take e = > 0. Since lim /(x) = A, for the chosen e > 0 there is 6 > 0 such that From relation (1) for the indicated values ​​of x we ​​have So, it has been found that, no matter how small, there are x Φ xQ, such that and at the same time ^ e. Hence the Definition. A function /(x) is said to be bounded in a neighborhood of the point x0 if there are numbers M > 0 and 6 > 0 such that Theorem 2 (boundedness of a function that has a limit). If the function f(x) is defined in a neighborhood of the point x0 and has a finite limit at the point x0, then it is bounded in some neighborhood of this point. m Let Then for any example, for e = 1, there is such 6 > 0 that for all x φ x0 satisfying the condition, the inequality will be true Noting that we always get Let. Then at each point x of the interval we have This means, according to the definition, that the function f(x) is bounded in a neighborhood. For example, the function /(x) = sin is bounded in a neighborhood of the point but has no limit at the point x = 0. Let us formulate two more theorems, the geometric meaning of which is quite clear. Theorem 3 (passing to the limit in inequality). If /(x) ⩽ ip(x) for all x in some neighborhood of the point x0, except perhaps for the point x0 itself, and each of the functions /(x) and ip(x) at the point x0 has a limit, then Note that that a strict inequality for functions does not necessarily imply a strict inequality for their limits. If these limits exist, then we can only assert that Thus, for example, the inequality while is true for functions. Theorem 4 (limit of an intermediate function). If for all x in some neighborhood of the point Xq, except, perhaps, the point x0 itself (Fig. 9), and the functions f(x) and ip(x) at the point xo have the same limit A, then the function f (x) at the point x0 has a limit equal to the same value of A. § ​​4. Limit of a function at infinity Let the function /(x) be defined either on the entire real axis or at least for all x satisfying the condition jx| > K for some K > 0. Definition. The number A is called the limit of the function f(x) as x tends to infinity, and they write if for any e > 0 there exists a number jV > 0 such that for all x satisfying the condition |x| > X, the inequality is true Replacing the condition in this definition accordingly, we obtain definitions From these definitions it follows that if and only if simultaneously That fact, geometrically means the following: no matter how narrow the e-strip between the lines y \u003d A- euy \u003d A + e, there is such a straight line x = N > 0 that to the right carried the graph of the function y = /(x) is entirely contained in the indicated e-strip (Fig. 10). In this case, they say that for x + oo, the graph of the function y \u003d / (x) asymptotically approaches the straight line y \u003d A. Example, The function / (x) \u003d jtjj- is defined on the entire real axis and is a fraction whose numerator is constant , and the denominator increases indefinitely as |x| +oo. It is natural to expect that lim /(x)=0. Let's show it. М Take any e > 0, subject to the condition For the relation to take place, the inequality c or must be satisfied, which is the same as whence Thus. if we take we will have. This means that the number is the limit of this function at Note that the radical expression is only for t ^ 1. In the case when, the inequality c is satisfied automatically for all Graph of an even function y = - asymptotically approaches the straight line Formulate using inequalities, which means §5. Infinitely Small Functions Let the function a(x) be defined in some neighborhood of the point x0, except possibly for the point x0 itself. Definition. The function a(x) is called an infinitesimal function (abbreviated as b.m.f.) as x tends to x0 if within the uniqueness of the limit boundedness of a function that has a limit transition to the limit in the inequality The limit of a function at infinity Infinitesimal functions Properties of infinitesimal functions For example, the function a(x) = x - 1 is b. m. f. at x 1, since lim (x-l) \u003d 0. The graph of the function y \u003d x-1 1-1 is shown in fig. II. In general, the function a(x)=x-x0 is the simplest example of b. m. f. at x-»ho. Taking into account the definition of the limit of a function at a point, the definition of b. m. f. can be formulated like this. Definition. A function a(x) is said to be infinitesimal for x - * xo if for any t > 0 there exists such "5 > 0 such that for all x satisfying the condition the inequality is true functions at Definition. The function a(x) is called infinitely small for x -» oo, if then the function a(x) is called infinitesimal, respectively, for or for For example, the function is infinitesimal for x -» oo, since lim j = 0. The function a (x ) = e~x is an infinitely small function as x - * + oo, since in what follows, we will, as a rule, consider all the concepts and theorems related to the limits of functions only in relation to the case of the limit of a function at a point, leaving the reader to formulate the corresponding concepts for himself and prove similar theorems of the day cases when Properties of infinitesimal functions Theorem 5. If a(x) and P(x) - b. m. f. for x - * xo, then their sum a(x) + P(x) is also a b.m. f. at x -» ho. 4 Take any e > 0. Since a(x) is a b.m.f. for x -* xo, then there is "51 > 0 such that for all x Φ xo satisfying the condition the inequality is true. By condition P(x) also b.m.f. for x ho, so there is such that for all χ φ ho satisfying the condition, the inequality is true Let us set 6 = min(«5j, 62). Then for all x Ф ho satisfying the condition, inequalities (1) and (2) will be simultaneously true. Therefore This means that the sum a(x) +/3(x) is a b.m.f. for xxq. Comment. The theorem remains valid for the sum of any finite number of functions, b. m. at x zo. Theorem 6 (product of a b.m.f. by a bounded function). If the function a(x) is b. m. f. for x -* x0, and the function f(x) is bounded in a neighborhood of the point Xo, then the product a(x)/(x) is 6. m. f. for x -» x0. By assumption, the function f(x) is bounded in a neighborhood of the point x0. This means that there are numbers 0 and M > 0 such that Let us take any e > 0. Since, by the condition, there is 62 > 0 such that for all x φ x0 satisfying the condition |x - xol, the inequality will be true Let i of all x f x0 satisfying the condition |x - x0|, the inequalities will be simultaneously true Therefore This means that the product a(x)/(x) is b. m.f. with Example. The function y \u003d xsin - (Fig. 12) can be considered as the product of the functions a (ar) \u003d x and f (x) \u003d sin j. The function a(a) is b. m. f. for x - 0, and the function f)