Let x – variable quantity. This means that the value x changes its meaning. This is what makes it fundamentally different from any constant value a, which does not change its unchanged value. For example, the height of a pole is a constant value, but the height of a living growing tree is a variable value.

Variable value x considered given, the numerical sequence is given

its meanings. That is, those values x 1 ; x 2 ;x 3 ;..., which it consistently, one after another, accepts in the process of its change. We will assume that this process of change by magnitude x its values ​​do not stop at any stage (variable X never freezes, it is “always alive”). This means that sequence (1) has an infinite number of values, which is marked in (1) by an ellipsis.

The values ​​of a variable can be considered as a set of values ​​of a function of the natural argument x n =f(n). Member x n is called the common member of the sequence. A sequence is considered given if a method is given for calculating any of its members by its known number.

Example 1: Write the first ten terms of the sequence if its common term is .

Solution: Calculating the value of a fraction given the values n equal to 1,2,3,…10, we get:

In general, a sequence with a common term will be written like this:

Naturally, interest arises regarding the nature of the change in magnitude x their meanings. That is, the question arises: do these values ​​change unsystematically, chaotically, or somehow purposefully?

The main interest is, of course, the second option. Namely, let the values x n variable x as their number increases n are approaching indefinitely ( strive) to some specific number a. This means that the difference (distance) between the values x n variable x and number a contracts, tending as it increases n(at ) to zero. Replacing the word “seeks” with an arrow, the above can be written as follows:

At<=>at (2)

If (2) holds, then we say that variable x tends to number a. This number A called limit of variable x. And it is written as follows:

Reads: limit x is a(x tends to a).

Aspiration variable x to your limit a can be clearly illustrated on a number axis. The exact mathematical meaning of this desire x To a is that no matter how small a positive number one takes, and therefore no matter how small the interval nor surround the number on the number line a, in this interval (in the so-called -neighborhood of the number a) will hit starting from a certain number N, all values x n variable x. In particular, in Fig. 1 into the depicted neighborhood of the number a got all the values x n variable x, starting from number .

Definition: Number A called the limit of the sequence (the limit of the variable X or the limit of the function f(n)), if whatever a predetermined positive number is, it is always possible to find such a natural number N, which for all members of the sequence with numbers n>N inequality will be satisfied.

This inequality is equivalent to the following two inequalities: . Number N depends on the chosen one. If you reduce the number , then the corresponding number N will increase.

For a sequence (or for a variable X) it is not necessary to have a limit, but if there is this limit, then it is the only one. A sequence having a limit is called convergent. A sequence that has no limit is called divergent.

Variable value x, can strive to its limit in various ways:

1. staying below your limit,

2. staying above your limit,

3. fluctuating around your limit,

4. taking values ​​equal to its limit.

The choice of number is arbitrary, but once it is chosen, it should not be subject to any further changes.

Variable x having zero as its limit (that is, tending to zero) is called infinitesimal. A variable x, growing without limit in absolute value is called infinitely large(its modulus tends to infinity).

So, if , then x is an infinitesimal variable quantity, and if , then x– an infinitely large variable quantity. In particular, if or , then x– an infinitely large variable quantity.

If , then . And vice versa if , That . From here we get the following important connection between the variable x and its limit a:

It has already been said that not every variable x has a limit. Many variables have no limit. Whether it exists or not depends on the sequence (1) of the values ​​of this variable.

Example 2 . Let

Here, obviously, that is.

Example 3 . Let

x– infinitesimal.

Example 4 . Let

Here, obviously, that is. So the variable x– infinitely large.

Example 5 . Let

Here, obviously, the variable x does not strive for anything. That is, it has no limit (does not exist).

Example 6 . Let

Here is the situation with the limit of the variable x not as obvious as in the previous four examples. To clarify this situation, let us transform the values x n variable x:

Obviously, at . Means,

at .

And this means that, that is.

Example 7 . Let

Here the sequence ( x n) variable values x represents an infinite geometric progression with the denominator q. Therefore, the limit of the variable x is the limit of an infinite geometric progression.

a) If , then, obviously, at . And this means that ().

b) If , then . That is, in this case the variable values x do not change - they are always equal to 1. Then its limit is also equal to 1 ().

c) If , then . In this case, it obviously doesn't exist.

d) If , then is an infinitely increasing positive number sequence. Which means ().

e) If , then introducing the notation , where , we obtain: – an alternating numerical sequence with terms infinitely increasing in absolute value:

Which means the variable x infinitely large. But due to the alternation of signs of its members, it tends neither to +∞ nor to –∞ (it has no limit).

Example 8. Prove that a sequence with a common term has a limit equal to 2.

Proof: Let us choose an arbitrarily positive number and show that it is possible to select such a number N, which for all values ​​of the number n, greater than this number N, the inequality will be satisfied, in which we must take a=2, , i.e. inequality will be satisfied .

From this inequality, after reduction in parentheses to a common denominator, we obtain . Thus: . Behind N Let's take the smallest integer belonging to the interval. Thus, we were able to determine, from an arbitrarily given positive, such a natural N that inequality performed for all numbers n>N. This proves that 2 is the limit of a sequence with a common term .

Of particular interest are monotonic and limited sequences.

Definition: monotonically increasing, if in front of everyone n each of its members is greater than the previous one, i.e. if , and monotonically decreasing if each of its terms is less than the previous one, i.e. .

Example 9. Sequence of natural numbers 1,2,3,…., n,… - monotonically increasing.

Example 10. Sequence of numbers, reciprocals of natural numbers, - monotonically decreasing.

Definition: the sequence is called limited, if all its terms are in a finite interval (-M,+M) And M>0, i.e. if , for any number n.

Example 11. Subsequence (xn), Where x n There is n the th decimal place of the number is limited, because .

Example 12. The sequence is limited because .

Basic properties of variables and their limits

1) If (variable x unchangeable and equal to constant a), then it is natural to assume that and . That is, the limit of a constant is equal to itself:

2) If , and a And b are finite, then . That is

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION STATE EDUCATIONAL INSTITUTION OF HIGHER PROFESSIONAL EDUCATION "NATIONAL RESEARCH TOMSK POLYTECHNIC UNIVERSITY" L.I. Samochernova HIGHER MATHEMATICS Part II Recommended as a textbook by the Editorial and Publishing Council of Tomsk Polytechnic University 2nd edition, revised Publishing House of Tomsk Polytechnic University 2005 UDC 514.12 C17 Samochernova L.I. C17 Higher mathematics. Part II: textbook / L.I. Samo-chernova; Tomsk Polytechnic University. – 2nd ed., rev. – Tomsk: Tomsk Polytechnic University Publishing House, 2005. – 164 p. The textbook includes three sections of higher mathematics: 1) introduction to mathematical analysis (the limit of a sequence and a function, infinitesimal and infinitely large quantities, comparison of infinitesimals, continuity of a function, discontinuity points); 2) differential calculus of a function of one variable (derivative and differential of a function, applications of differential calculus to the study of functions); 3) integral calculus (indefinite integral, definite integral, geometric applications of definite integral). The manual was prepared at the Department of Applied Mathematics and is intended for students of foreign education studying in the areas 080400 “Human Resources Management”, 080200 “Management”, 080100 “Economics”, 100700 “Trading”. UDC 514.12 Reviewers Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Algebra of TSU S.Ya. Grinshpon Candidate of Technical Sciences, Associate Professor of the Faculty of Control Systems of TUSUR A.I. Kochegurov © Tomsk Polytechnic University, 2005 © Samochernova L.I., 2005 © Design. Tomsk Polytechnic University Publishing House, 2005 2 1. INTRODUCTION TO MATHEMATICAL ANALYSIS 1.1. A numerical sequence and its limit Definition 1. If, according to some law, each natural number n is associated with a well-defined number xn, then we say that a numerical sequence (xn) is given: x1,x2, x3,...,xn,. .. (1.1) In other words, a number sequence is a function of a natural argument: xn = f(n). The numbers that make up a sequence are called its terms, and xn is the common or nth term of the sequence. Example of a number sequence: 2, 4, 6, 8, ..., 2n, ... For this sequence x1 = 2, x2 = 4, x3 = 6,..., x n = 2n is a common member of the sequence of even numbers. n Example 1. Knowing the general term of the sequence xn = , write n+2 its first five terms. Solution. Giving n the values ​​1, 2, 3, 4, 5, we get 1 2 3 4 5 x1 = ; x2 = ; x3 = ; x4 = ; x5 = . 3 4 5 6 7 n In general, a sequence with a common term xn = will be written like this: n+2 1 2 3 4 n ,...,... 3 4 5 6 n+2 Note that since xn =f(n) is a function, that is, generally speaking, a variable quantity, then for convenience we will often call the function xn a variable quantity, or simply a variable xn. Bounded and unbounded sequences Definition 2. A sequence (xn) is called bounded from above (from below) if there is a real number M (number m) such that each element xn of the sequence (xn) satisfies the inequality xn ≤ M (xn ≥ m). In this case, the number M (number m) is called the upper bound (lower bound) of the sequence (xn), and the inequality xn ≤ M (xn ≥ m) is called the condition for the sequence to be bounded from above (from below). 3 Definition 3. A sequence is called bounded on both sides, or simply bounded, if it is bounded both above and below, that is, if there are numbers m and M such that any element xn of this sequence satisfies the inequalities: m ≤ xn ≤ M. If a sequence (xn) is bounded and M and m are its upper and lower bounds, then all elements of this sequence satisfy the inequality xn ≤ A, (1.2) where A is the maximum of two numbers |M| and |m|. Conversely, if all elements of the sequence (xn) satisfy inequality (1.2), then the inequalities − A ≤ xn ≤ A also hold and, therefore, the sequence (xn) is bounded. Thus, inequality (1.2) is another form of the condition for the boundedness of a sequence. Let us clarify the concept of an unbounded sequence. A sequence (xn) is called unbounded if for any positive number A there is an element xn of this sequence that satisfies the inequality xn > A. 2n Examples: 1. A sequence with a common term xn = (− 1)n sin 3n n +1 is bounded, since for all n the inequality 2n 2n xn = (− 1)n ⋅ ⋅ sin 3n ≤< 2 (A = 2). n +1 n +1 2. Последовательность 1, 2, 3, 4, ..., n, ..., общий член которой xn = n , очевидно, неограниченная. В самом деле, каково бы ни было положительное число А, среди элементов этой последовательности найдутся элементы, пре- восходящие А. Монотонные последовательности Определение 4. Последовательность {xn } называется неубывающей (невозрастающей), если каждый последующий член этой последовательно- сти не меньше (не больше) предыдущего, то есть для всех номеров n спра- ведливо неравенство xn ≤ xn +1 (xn ≥ xn +1) . Неубывающие и невозрастающие последовательности объединяются общим наименованием монотонные последовательности. Если элементы монотонной последовательности {xn } для всех номеров n удовлетворяют не- равенству xn < xn +1 (xn >xn +1), then the sequence (xn) is called increasing (decreasing). Increasing and decreasing sequences are also called strictly monotonic. Example 2. The sequence of odd numbers 1, 3, 5, 7, ..., 2n–1, ..., where xn = 2n − 1, is monotonically increasing. 4 Indeed, xn +1 − xn = − (2n − 1) = 2, so xn +1 − xn > 0, i.e. xn +1 > xn for all n. Limit of a sequence Let us define one of the most important concepts of mathematical analysis - the limit of a sequence, or, what is the same, the limit of a variable xn running through the sequence x1,x2,...,xn,... Definition 5. A constant number a is called a limit sequence x1,x2 ,...,xn,... or the limit of the variable xn, if for any arbitrarily small positive number ε one can specify a natural number N such that for all members of the sequence with numbers n>N you - the inequality xn − a is fulfilled< ε. (1.3) Тот факт, что последовательность (1.1) имеет своим пределом число а, обо- значается так: lim xn = a или xn → a ; n→∞ n→∞ (lim есть сокращённое обозначение латинского слова limes, означающего «предел»). Последовательность, имеющую пределом число а, иначе называют по- следовательностью, сходящейся к а. Последовательность, не имеющая пре- дела, называется расходящейся. Замечание. Величина N зависит от ε, которое мы выбираем произволь- ным образом (N=N(ε)). Чем меньше ε, тем N, вообще говоря, будет больше. Исключением является случай, когда последовательность состоит из одина- ковых членов. 1 2 3 n Пример 3. Доказать, что последовательность, L,L 2 3 4 n +1 n с общим членом xn = имеет предел, равный 1. n +1 Решение. Выберем произвольно положительное число ε и покажем, что для него можно найти такое натуральное число N, что для всех номеров n >N inequality (1.3) will be satisfied, in which we must take a =1; n xn = , that is, the inequality n +1 n 1−< ε. (1.4) n +1 После приведения к общему знаменателю в левой части неравенства (1.4) получим 5 n +1− n 1 < ε или < ε. n +1 n +1 Но если 1 /(n + 1) < ε, то и 1 /(n + 1) < ε. Из последнего неравенства следу- ет, что n + 1 >1/ε, n > 1/ε–1. Consequently, N can be taken to be the largest integer contained in (1/ε – 1), that is, E(1/ε – 1). Then inequality (1.4) will be satisfied for all n >N. If it turns out that E(1/ε – 1) ≤ 0, then N can be taken equal to 1. Since ε was taken arbitrarily, this proves that 1 is the limit of the sequence with a common term xn = n /(n + 1) . In particular, if ε = 0.01, then N = E (1 / 0.01 − 1) = E (100 – 1) = 99; if ε=1/2, then N=E (1 / 0.5 − 1)=1, etc. N chosen in this way for different values ​​of ε will be the smallest possible. Geometric interpretation of the limit of a number sequence The number sequence (1.1) can be considered as a sequence of points on a line. In the same way, we can talk about a limit as a point on a line. Since the inequality xn − a< ε равносильно неравенству – ε < xn − a < ε, которое, в свою очередь, равносильно такому a – ε < xn < a + ε, то определение предела числовой последовательности можно сформулировать и так. Определение 6. Точка а называется пределом последовательности то- чек (1.1), если, какую бы окрестность (a – ε, a + ε) точки а мы ни задали, найдётся такое число N, что все точки последовательности (1.1) с номерами n >N will fall into the given neighborhood. Let us represent the numbers a, a – ε, a + ε and the values ​​of the variable xn as points on the number axis (Fig. 1). The fulfillment of inequality (1.3) under the condition n > N geometrically means that all points xn, starting from the point x N +1, that is, from a point whose index exceeds some natural number N, will certainly lie in the ε-neighborhood points a. Outside this neighborhood, even if there are points xn, there will be only a finite number of them. Rice. 1 Test for the convergence of a monotone sequence Theorem 1. Every non-increasing (non-decreasing) sequence (xn) or variable xn bounded from below (from above) has a limit. 6 1.2. Infinitely small and infinitely large quantities Definition 1. A variable xn is called infinitesimal if it has a limit equal to zero. Following the definition of the limit, we can say that xn will be infinitesimal if for any arbitrarily small ε > 0 there is N such that for all n > N the inequality xn< ε. Иначе говоря, бесконечно малой называется такая переменная величина xn , которая при своём изменении, на- чиная с некоторого номера n, становится и остаётся по абсолютной величине меньше любого наперёд заданного числа ε >0. Examples of an infinitesimal are the variables 1 1 (−1) n xn = , xn = − , xn = , xn = q n for q< 1 и другие. n n n Пример 1. Доказать, что xn = (− 1)n есть бесконечно малая. n Решение. (−1) n 1 Возьмем произвольное ε >0. From the inequality xn = =< ε полу- n n чаем n >1/ε. If we take N = E(1/ε), then for n > N we will have xn< ε. При 1 ε= получим N = E(10) = 10, при ε = 4 / 15 получим N = E (15 / 4) = 3 и т. д. 10 А это и значит, что xn = (− 1)n есть бесконечно малая. n Замечание 1. Нельзя смешивать постоянное очень малое число с бес- конечно малой величиной. Единственным числом, которое рассматривается в качестве бесконечно малой величины, служит нуль (в силу того, что предел постоянной равен ей самой). Определение 2. Переменная величина xn называется бесконечно большой величиной, если для любого наперед заданного сколь угодно боль- шого числа M >0 we can specify a natural number N such that for all numbers n > N the inequality xn > M holds. In other words, a variable xn is called infinitely large if, starting from a certain number, it becomes and remains for all subsequent numbers are greater in absolute value than any predetermined positive number M. An infinitely large variable xn is said to tend to infinity or has an infinite limit, and they write: xn → ∞ or lim xn = ∞. n →∞ n →∞ 7 In connection with the introduction of a new concept - “infinite limit” - we agree to call a limit in the previously defined sense a finite limit. Example 2. The quantity xn = (− 1)n ⋅ n, taking successively the values ​​-1, 2, -3, 4, -5, ..., (− 1)n n, K is infinitely large. Indeed, xn = (− 1)n n = n . From here it is clear that, whatever the number M, for all n, starting from some, there will be xn = n > M, that is, lim xn = ∞. n →∞ Definition 3. A variable quantity xn is called a positive infinitely large quantity if for any number M one can specify a natural number N such that for all numbers n > N the inequality xn > M holds. In this case, the variable quantity is said to be xn tends to plus infinity and symbolically write it like this: xn → +∞ or lim xn = +∞. n→∞ n →∞ Definition 4. A variable xn is called a negative infinitely large quantity if for any number M one can specify a natural number N such that for all n > N the inequality xn holds<М. В этом случае говорят, что переменная величина xn стремится к минус бесконечности и записывают это так: xn → −∞ или lim xn = −∞ . n→∞ n →∞ Так, например, xn = n будет положительной, а xn = −n – отрицательной бесконечно большой величиной. Переводя предыдущие определения на геометрический язык, мы можем сказать: если xn – бесконечно большая величина, то, как бы ни был велик сегмент длины 2М (М > 0) with the center at the origin of coordinates, the point xn, representing the values ​​of an infinitely large quantity, with a sufficiently large number n will be outside the indicated segment and with a further increase in n will remain outside it (Fig. 2). Moreover, if xn is a positive (negative) infinitely large quantity, then the point representing its values ​​will be for sufficiently large numbers n outside the specified segment on the right (left) side of the origin. Rice. 2 8 Remark 2. 1. The symbols ∞, + ∞, − ∞ are not numbers, but are introduced only to simplify notation and to briefly express the fact that a variable is infinitely large, positive infinitely large and negative infinitely large. It should be firmly remembered that no arithmetic operations can be performed on these symbols! 2. You cannot mix a constant very large number with an infinitely large value. Relationship between infinitely large and infinitesimal quantities Theorem 1. Let xn ≠0 (for any n). If xn is infinitely large, then yn = 1 / xn is infinitely small; if xn is infinitely small, then yn = 1 / xn is infinitely large. 1.3. Arithmetic operations on variable quantities. Basic theorems on the limits of variables (sequences) Let us introduce the concept of arithmetic operations on variables. Let us have two variable quantities xn and yn, taking the following values, respectively: x1, x2, x3, ..., xn, ..., y1, y2, y3, ..., yn, .... The sum of two given variables xn and yn is understood as a variable, each value of which is equal to the sum of the corresponding (with the same numbers) values ​​of the variables xn and yn, that is, a variable taking a sequence of values ​​x1 + y1, x2 + y2, K, xn + yn , K We will denote this variable by xn + yn . The sum of any number of variables, their product, as well as the difference of two variables and their quotient are determined similarly. Thus, new variables arise: xn + y n, xn − y n, xn ⋅ y n and x n / y n. (In the latter case, it is assumed that, at least from some number, yn ≠0, and the quotient xn / yn is considered only for such numbers). Similarly, these definitions are formulated in terms of sequences. 9 Theorems on the limits of variables Theorem 1. The variable xn can have only one limit. There is a connection between variables that have a limit and infinitesimal quantities. Theorem 2. A variable quantity that has a limit can be represented as the sum of its limit and some infinitesimal quantity. Theorem 3 (converse to Theorem 2). If the variable xn can be represented as the sum of two terms xn = a + α n, (1.5) where a is a certain number and α n is an infinitesimal, then a is the limit of the variable xn. Theorem 4. If a variable xn has a finite limit, then it is bounded. Consequence. An infinitesimal variable is limited. Lemma 1. The algebraic sum of any (but limited) number of infinitesimal quantities is also an infinitesimal quantity. Lemma 2. The product of a bounded variable xn and an infinitesimal α n is an infinitesimal quantity. Corollary 1. The product of any finite number of infinitesimal quantities represents an infinitesimal quantity. Corollary 2. The product of a constant quantity and an infinitesimal quantity is an infinitesimal quantity. Corollary 3. The product of a variable quantity tending to the limit and an infinitesimal quantity is an infinitesimal quantity. Using Lemmas 1 and 2, we can prove the following theorems about limits. Theorem 5. If the variables xn and yn have finite limits, then their sum, difference, product also have finite limits, and: 1) lim (xn ± yn) = lim xn ± lim yn , n→∞ n→∞ n→∞ 2) lim (xn ⋅ yn) = lim xn ⋅ lim yn. n→∞ n→∞ n→∞ Remark 1. This theorem is true for any fixed number of terms and factors. Consequence. The constant factor can be taken beyond the sign of the limit, i.e. lim (cxn) = c lim xn , n →∞ n→∞ where c is some constant. Theorem 6. If the variables xn and yn have finite limits and yn ≠0, lim yn ≠ 0, then the quotient of these variables also has a limit, and n →∞ 10

Number sequence.

Variable running through a number sequence

If for every natural number n assigned a real number x n, i.e.

1, 2, 3, 4, …, n, …

x 1 , x 2 , x 3 , x 4 , …, x n , …

then they say that a number sequence with a common term is given x n. In what follows we will say that the variable is given x, running through a number sequence with a common term x n. In this case, we will denote this variable x n. Variable values x n are represented by points on the number axis.

For example, given the variables:

: or ;

: 1, 4, 6, …, 2n ..

Number A called limit of variable x n , if for any arbitrarily small number ε > 0 there is such a natural number N x n, whose number n more number N, satisfy the inequality.

This fact is symbolically written as follows:

Geometrically, this means that the points representing the values ​​of the variable x n, thicken, accumulate around a point A.

Note that if a variable has a limit, then it is the only one. The limit of a constant is the constant itself, i.e. , If c=const. A variable may not have a limit at all.

For example, variable x n =(-1) n has no limit, i.e. There is no single number around which the values ​​of a variable accumulate. Geometrically it's obvious .

Restricted variable

Variable x n called limited , if such a number exists M> 0, which | x n| < M for all numbers n.

Given a variable. As a number M you can take, for example, 3. Obviously, for all numbers n. Therefore, is a limited variable.

Variable x n = 2n is unlimited, because as the number increases n its values ​​increase and it is impossible to find such a number M> 0 to |2 n| < M for all numbers n.

Theorem. If a variable has a finite limit, then it is limited.

The converse theorem is not true.

Infinitesimal quantities

Variable x n called infinitesimal , if its limit is 0.

For example, the quantities that are infinitesimal are:

Because ;

Because

The quantity is not infinitesimal, it is a finite quantity.

The sum (difference) of a finite number of infinitesimals is an infinitesimal quantity.

The product of an infinitesimal by a constant quantity or by an infinitesimal or by a quantity having a finite limit is an infinitesimal quantity.

Infinitely large quantities

Variable x n called infinitely large , if for any arbitrarily large number A>0, there is such a natural number N, that all values ​​of the variable x n, whose number n>N, satisfy the inequality.

In this case they write or.

For example, the following variables are infinitely large:

x n = n 2 : 1,4,9,16,…; x n = -5n: -5, -10, -15, -20, …;

x n = (-1) n ×n: -1, 2, -3, 4, -5, 6, … .

It can be seen that the magnitudes of the values ​​of these variables increase without limit.

, , .

The product of an infinitely large by an infinitely large or a quantity having a limit is an infinitely large quantity.

The sum of infinitely large numbers of one sign is infinitely large.

The reciprocal of infinitely large is infinitesimal.

The reciprocal of an infinitesimal is an infinitely large.

Comment.

If , A- number, then they say that x n It has finite limit.

If , then they say that x n It has endless limit.

Arithmetic operations on variables

If the variables x n And y n have finite limits, then their sum, difference, product and quotient also have finite limits, and if and then

(4.3)

Comment: , c = const.

The constant factor can be taken beyond the limit sign.

Function

Let two variables be given x And y.

Variable y called function from variable x, if each value x from a certain set, according to a certain law, a certain value corresponds y.

Wherein x called independent variable or argument , y – dependent variable or function . Indicated by: y = f(x) or y=y(x).

Limit is one of the most fundamental concepts of higher mathematics. In this chapter we will look at two main types of limits: 1) limit of a variable; 2) limit of the function.

Let X – Variable value. This means that the value X changes its meaning. This is what makes it fundamentally different from any Constant value A, which does not change its unchanged value. For example, the height of a pole is a constant value, but the height of a living growing tree is a variable value.

Variable value X is considered given if the sequence is given

Its meanings. That is, those values X 1; X 2; X 3;..., which it consistently, one after another, accepts in the process of its change. We will assume that this process of change by magnitude X its values ​​do not stop at any stage (variable X never freezes, it is “always alive”). This means that sequence (1.1) has an infinite number of values, which is marked in (1.1) by an ellipsis.

Naturally, interest arises regarding the nature of the change in magnitude X their meanings. That is, the question arises: do these values ​​change unsystematically, chaotically, or somehow purposefully?

The main interest is, of course, the second option. Namely, let the values Xn Variable X as their number increases N are approaching indefinitely ( strive) to some specific number A. This means that the difference (distance) between the values Xn Variable X and number A contracts, tending as it increases N(at ) to zero. Replacing the word “seeks” with an arrow, the above can be written as follows:

At<=>at (1.2)

If (1.2) holds, then we say that The variable x tends to the number a. This number A Called Limit of variableX. And it is written as follows:

<=> (1.3)

Reads: LimitXequalsA (Xstrives forA).

Aspiration variable X to your limit A Can be clearly illustrated on a number axis. The exact mathematical meaning of this desire X To A is that no matter how small a positive number one takes, and therefore no matter how small the interval nor surround the number on the number line A, in this interval (in the so-called -neighborhood of the number A) will hit starting from a certain number N, all values Xn Variable X. In particular, in Fig. 3.1 in the depicted neighborhood of the number A got all the values Xn Variable X, starting from number .

Variable X having zero as its limit (that is, tending to zero) is called Infinitesimal. A variable X, growing without limit in absolute value is called Infinitely large(its modulus tends to infinity).

So, if , then X is an infinitesimal variable quantity, and if , then X– an infinitely large variable quantity. In particular, if or , then X– an infinitely large variable quantity.

If , then . And vice versa if , That . From here we get the following important connection between the variable X and its limit A:

Note that not every variable X has a limit. Many variables have no limit. Whether it exists or not depends on what the sequence (1.1) of the values ​​of this variable is.

Example 1 . Let

Here, obviously, that is.

Example 2 . Let

X– infinitesimal.

Example 3 . Let

Here, obviously, that is. So the variable X– infinitely large.

Example 4 . Let

Here, obviously, the variable X does not strive for anything. That is, it has no limit (does not exist).

Example 5 . Let

Here is the situation with the limit of the variable X not as obvious as in the previous four examples. To clarify this situation, let us transform the values Xn variable X:

Obviously, at . Means,

at .

And this means that, that is.

Example 6 . Let

Here the sequence ( Xn) variable values X represents an infinite geometric progression with the denominator Q. Therefore, the limit of the variable X is the limit of an infinite geometric progression.

A) If , then, obviously, at . And this means that ().