Limit of a function – definitions, theorems and properties. Infinitesimal functions

The concept of limit and the concept of function are fundamental concepts mathematical analysis. The study of the concept of a limit began in elementary mathematics, where the length of a circle, the volume of a cylinder, a cone, etc. are determined using passages to the limit. It was also used in determining the sum of an infinitely decreasing geometric progression. The operation of passing to the limit is one of the main operations of analysis.

Purpose of function (limit value of function) V given point, limiting for the domain of definition of a function, is the value to which the value of the function under consideration tends as its argument tends to a given point.

Function limit is a generalization of the concept of a limit of a sequence: initially, the limit of a function at a point was understood as the limit of a sequence of elements of the domain of values of a function composed of images of points of a sequence of elements of the domain of definition of a function converging to a given point (the limit at which is considered); if such a limit exists, then the function is said to converge to the specified value; if such a limit does not exist, then the function is said to diverge.

Most common definition function limit formulated in the language of the surrounding area. The fact that the limit of a function is considered only at points that are limiting for the domain of definition of the function means that in each neighborhood of a given point there are points of the domain of definition; this allows us to talk about the tendency of the function argument (towards a given point). But the limit point of the domain of definition does not have to belong to the domain of definition itself: for example, one can consider the limit of a function at the ends of the open interval on which the function is defined (the ends of the interval themselves are not included in the domain of definition).

IN general case it is necessary to accurately indicate the method of convergence of the function, for which the so-called a base of subsets of the domain of definition of the function, and then the definition of the limit of the function from the (given) base is formulated. In this sense, the system of punctured neighborhoods of a given point is special case such a base of sets.

Since on the extended real line it is possible to construct a base of neighborhoods of a point at infinity, it turns out to be possible to describe the limit of a function when the argument tends to infinity, as well as to describe the situation when the function itself tends to infinity (at a given point). The limit of the sequence (as the limit of the function of the natural argument) provides an example of convergence over the base “the tendency of the argument to infinity.”

The absence of a limit of a function (at a given point) means that for any predetermined value of the range of values there is a neighborhood of this value such that in any arbitrarily small neighborhood of the point at which the function takes set value, there are points at which the value of the function will be outside the specified neighborhood.

If at some point in the domain of definition of a function there is a limit and this limit is equal to the value of the function at a given point, then the function turns out to be continuous (at a given point).

Let the function is defined in some neighborhood of point a, except, perhaps, the point a itself.

The number B is called the limit of the function at point a (or at ), if for any sequence of argument values , the sequence of corresponding function values, converges to the number B

The number A is called the limit of the function at the point x=x0(or for), if for any convergent to x0 sequence (1) argument value x, different from x0, the corresponding sequence (2) of function values converges to the number A. Indicated.

The function may have at a point x0 only one limit. This follows from the fact that the sequence has only one limit.

1). Function =с=const has a limit at every point x0 number line, i.e.

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2). Function = x has at any point x0 number line limit equal to x0, i.e.

Definition 2. Number A is called the limit of the function at the point x=x0, if for any number there is a number such that for all satisfying the inequality the inequality holds. limit function numerical factor

The first definition is based on the concept of the limit of a number sequence, so it is often called the definition “in the language of sequences”, or the definition according to Heine (1821-1881 - German mathematician). The second definition is called the definition in the language "", or the definition according to Cauchy (1789-1857 - French mathematician).

It can be proven that both definitions of the limit of a function at a point x0 are equivalent, which means that you can use any of them depending on which one is more convenient when solving a particular problem.

In addition to the considered concept of the limit of a function at, there is also the concept of the limit of a function at.

Definition. Number A is called the limit of a function at if for any E>0 you can specify such a positive number N, which for all values x, satisfying the inequality, the inequality will be satisfied.

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The intuitive concept of limiting passage was used by scientists Ancient Greece when calculating the areas and volumes of various geometric shapes. Methods for solving such problems were mainly developed by Archimedes.

When creating differential and integral calculus, mathematicians of the 17th century (and, above all, Newton) also explicitly or implicitly used the concept of passage to the limit. The definition of the concept of limit was first introduced in the work of Wallis "Arithmetic of Infinite Quantities"(XVII century), however, historically this concept did not form the basis of differential and integral calculus.

With the help of the theory of limits in the second half of the 19th century, in particular, the use of infinite series in analysis was justified, which were a convenient apparatus for constructing new functions.

Sequence limit

Main article: Sequence limit

Number a (\displaystyle a) called the limit of the sequence a n = ( x 1 , x 2 , . . . , x n ) (\displaystyle a_(n)=\(x_(1),x_(2),...,x_(n)\)) , If ϵ > 0 (\displaystyle \epsilon >0) , ∃ (\displaystyle \exists ) N (ϵ) (\displaystyle N(\epsilon)) , ∀ (\displaystyle \forall ) n > N (ϵ) (\displaystyle n>N(\epsilon)): | a n − a |< ϵ {\displaystyle |a_{n}-a|<\epsilon } . The sequence limit is indicated by lim n → + ∞ a n (\displaystyle \lim _(n\to +\infty )a_(n)). Where exactly is it aiming? n (\displaystyle n), may not be specified, since n (\displaystyle n) ∈ N (\displaystyle \in \mathbb (N) ), it can only tend to + ∞ (\displaystyle +\infty ).

Properties:

If a sequence limit exists, then it is unique.
lim c = c (\displaystyle \lim c=c) , c − c o n s t (\displaystyle ,c-const)
lim (x n + y n) = lim x n + lim y n (\displaystyle \lim(x_(n)+y_(n))=\lim x_(n)+\lim y_(n))
lim (q x n) = q lim x n (\displaystyle \lim(qx_(n))=q\lim x_(n)) , q − c o n s t (\displaystyle ,q-const)
lim (x n y n) = lim x n lim y n (\displaystyle \lim(x_(n)y_(n))=\lim x_(n)\lim y_(n))(if both limits exist)
lim (x n / y n) = lim x n / lim y n (\displaystyle \lim(x_(n)/y_(n))=\lim x_(n)/\lim y_(n))(if both limits exist and the denominator of the right-hand side is not zero)
If a n > x n > b n ∀ n (\displaystyle a_(n)>x_(n)>b_(n)\forall n) And lim a n = lim b n (\displaystyle \lim a_(n)=\lim b_(n)), That lim x n = lim a n = lim b n (\displaystyle \lim x_(n)=\lim a_(n)=\lim b_(n))(the "sandwiched sequence theorem", also known as the "two policemen theorem")

Function limit

Main article: Limit of a function

A number b is called the limit of the function f(x) at point a if ∀ ϵ > 0 (\displaystyle \forall \epsilon >0) exists δ > 0 (\displaystyle \delta >0), such that ∀ x , 0< | x − a | < δ {\displaystyle \forall x,0<|x-a|<\delta } running | f (x) − b |< ϵ {\displaystyle |f(x)-b|<\epsilon } .

For limits of functions, similar properties are valid as for limits of sequences, for example, lim x → x 0 (f (x) + g (x)) = lim x → x 0 f (x) + lim x → x 0 g (x) (\displaystyle \lim _(x\to x_(0) )(f(x)+g(x))=\lim _(x\to x_(0))f(x)+\lim _(x\to x_(0))g(x)), if all members exist.

Generalized concept of the limit of a sequence

Let X (\displaystyle X)- a set in which the concept of neighborhood is defined U (\displaystyle U)(for example, metric space). Let x i ∈ X (\displaystyle x_(i)\in X)- a sequence of points (elements) of this space. They say that x ∈ X (\displaystyle x\in X) there is a limit of this sequence if in any neighborhood of the point x (\displaystyle x) almost all terms of the sequence lie, that is ∀ U (x) ∃ n ∀ i > n x i ∈ U (x) (\displaystyle \forall U(x)\exists n\forall i>nx_(i)\in U(x))

Limits give all mathematics students a lot of trouble. To solve a limit, sometimes you have to use a lot of tricks and choose from a variety of solution methods exactly the one that is suitable for a particular example.

In this article we will not help you understand the limits of your capabilities or comprehend the limits of control, but we will try to answer the question: how to understand limits in higher mathematics? Understanding comes with experience, so at the same time we will give several detailed examples of solving limits with explanations.

The concept of limit in mathematics

The first question is: what is this limit and the limit of what? We can talk about the limits of numerical sequences and functions. We are interested in the concept of the limit of a function, since this is what students most often encounter. But first, the most general definition of a limit:

Let's say there is some variable value. If this value in the process of change unlimitedly approaches a certain number a , That a – the limit of this value.

For a function defined in a certain interval f(x)=y such a number is called a limit A , to which the function tends when X , tending to a certain point A . Dot A belongs to the interval on which the function is defined.

It sounds cumbersome, but it is written very simply:

Lim- from English limit- limit.

There is also a geometric explanation for determining the limit, but here we will not delve into the theory, since we are more interested in the practical rather than the theoretical side of the issue. When we say that X tends to some value, this means that the variable does not take on the value of a number, but approaches it infinitely close.

Let's give a specific example. The task is to find the limit.

To solve this example, we substitute the value x=3 into a function. We get:

By the way, if you are interested, read a separate article on this topic.

In examples X can tend to any value. It can be any number or infinity. Here's an example when X tends to infinity:

Intuitively, the larger the number in the denominator, the smaller the value the function will take. So, with unlimited growth X meaning 1/x will decrease and approach zero.

As you can see, to solve the limit, you just need to substitute the value to strive for into the function X . However, this is the simplest case. Often finding the limit is not so obvious. Within the limits there are uncertainties of the type 0/0 or infinity/infinity . What to do in such cases? Resort to tricks!

Uncertainties within

Uncertainty of the form infinity/infinity

Let there be a limit:

If we try to substitute infinity into the function, we will get infinity in both the numerator and the denominator. In general, it is worth saying that there is a certain element of art in resolving such uncertainties: you need to notice how you can transform the function in such a way that the uncertainty goes away. In our case, we divide the numerator and denominator by X in the senior degree. What will happen?

From the example already discussed above, we know that terms containing x in the denominator will tend to zero. Then the solution to the limit is:

To resolve type uncertainties infinity/infinity divide the numerator and denominator by X to the highest degree.

By the way! For our readers there is now a 10% discount on

Another type of uncertainty: 0/0

As always, substituting values into the function x=-1 gives 0 in the numerator and denominator. Look a little more closely and you will notice that we have a quadratic equation in the numerator. Let's find the roots and write:

Let's reduce and get:

So, if you are faced with type uncertainty 0/0 – factor the numerator and denominator.

To make it easier for you to solve examples, we present a table with the limits of some functions:

L'Hopital's rule within

Another powerful way to eliminate both types of uncertainty. What is the essence of the method?

If there is uncertainty in the limit, take the derivative of the numerator and denominator until the uncertainty disappears.

L'Hopital's rule looks like this:

Important point : the limit in which the derivatives of the numerator and denominator stand instead of the numerator and denominator must exist.

And now - a real example:

There is typical uncertainty 0/0 . Let's take the derivatives of the numerator and denominator:

Voila, uncertainty is resolved quickly and elegantly.

We hope that you will be able to usefully apply this information in practice and find the answer to the question “how to solve limits in higher mathematics.” If you need to calculate the limit of a sequence or the limit of a function at a point, and there is absolutely no time for this work, contact us for a quick and detailed solution.

The origin and creation of the theory of real numbers

3 Formation of limit theory

A rigorous mathematical construction of the concept of a real number became possible thanks to the theory of limits.

A person who has received a modern mathematical education has difficulty imagining differential and integral calculus without the apparatus of limit theory. However, historically the derivative appeared before the limit. The reasons for this phenomenon are explained by the urgent need of natural science in the 17th century for methods of differential and integral calculus.

In the 17th century, ideas related to infinitesimal methods began to develop rapidly. Here it is worth noting such mathematicians as Descartes, Fermat, Pascal, Torricelli, Cavalieri, Roberval, Barrow. The quadrature method, developed in antiquity, has found wide application and development. The question of tangents was studied - a definition was given that was more general than the ancient one, and methods for finding tangents were constructed. Attempts have been made to introduce a derivative. It was even found that the problem of finding a tangent is the inverse of the problem of quadrature.

Despite the lack of rigor, “...mathematicians became increasingly skilled in handling the concepts underlying infinitesimal calculus.”

Infinitesimal methods are gaining popularity among mathematicians and are being increasingly used and improved. Integral and differential calculus is gradually formalized and generalized by the works of such scientists as Newton (1643-1727) and Leibniz (1646-1716). Thus, Newton established a connection between the derivative and the integral and proposed a new method for solving equations using the derivative. He developed the fluxion method, which related the derivative to instantaneous velocity and acceleration. Using this method, he developed integral and differential calculus. Newton also proposed an algorithm for finding the derivative of a function, based on an early form of limit theory. The basis and powerful tool of the fluxion method was the expansion of functions into series, although without proper justification for their convergence.

We owe Leibniz a large number of convenient and beautiful notations in integral and differential calculus. Leibniz arrived at his results independently of Newton. Using knowledge from combinatorics, he developed a formal method for calculating integrals. Leibniz introduced the concept of differential by defining it in terms of tangents, found some rules for finding the differential of a complex function, and also introduced differentials of higher orders. Leibniz also developed methods for searching for extremum points and inflection points. The strength of Leibniz's theory, from the point of view of practical calculations, was its algorithmic nature and formality.

Both Newton and Leibniz solved many practically important problems using the concepts of infinitesimal quantities; their points of view on the derivative and integral differed from each other. Thus, Newton uses the fluxion method to solve differential problems, and Leibniz uses differentials. Newton considers integration as the inverse problem of differentiation (in our terms, finding the antiderivative), and Leibniz considers the integral as the sum of the areas of infinitesimal rectangles. It is quite natural that these two concepts were competing with each other.

Newton and Leibniz, using infinitesimals in their calculations, could not explain their nature, because they did not imagine a small quantity that was both finite and different from 0. Both scientists came close to the concept of limit, but “...a narrow concept of number that did not allow identification certain relations with numbers, was partly the reason that the concept of a limit could not “erupt” in either Newton’s or Leibniz’s theories.” Mathematicians used intuitive and geometric considerations. Functions were understood as curves obtained by some movement (just as they were considered by the ancient Greeks). “The first creators of analysis and their followers took for granted the validity of two basic ideas about space and mechanical motion.” It is probably for this reason that the connection between continuity and differentiability has long been considered almost synonymous.

However, the infinitesimal method proved its fruitfulness and necessity in mathematics, which made the problem of the foundation for integral and differential calculus even more acute. The debate was not only among mathematicians; All mathematics was subjected to severe attacks, for example, from the theologian D. Berkeley. This state of mathematics XVII-XVII was called the second crisis of mathematics.

Following Newton and Leibniz, attempts to define the concept of the infinitesimal were made by Euler, d'Alembert and Lagrange. These attempts cannot be called useless; these works strengthened the concept of functions in mathematics, which played a role in the further search for the theory of the limit. However, it was not possible to construct a coherent and logically substantiated theory.

Thus, by the 19th century a paradoxical situation had developed in mathematics. There were undoubted successes of mathematical sciences in natural science, a methodology for treating series, differentiation and integration was developed, many important problems were solved, but there was no understanding of what mathematical analysis was based on. The need to understand the foundations of the new mathematics has become universal and urgent.

We owe the construction of a harmonious and rigorous theory of infinitesimals to Augustin Louis Cauchy (1789-1857). It should be recognized that Cauchy was not the first mathematician to come up with this idea, but, historically, his work played a key role in the development of mathematical analysis. Cauchy gave a general definition of a limit in descriptive form: “If the values successively assigned to the same variable approach a fixed value indefinitely, so that in the end they differ from it as little as possible, then the latter is called the limit of all others.” Quote taken from. From the point of view of this definition, it became clear that an infinitesimal quantity is just a quantity that has a limit equal to 0, then Cauchy defined the concept of a derivative and showed the connection of this definition with Leibniz differentials. He also built the first rigorous theory of integration and proved the connection between integration and differentiation.

It is difficult to overestimate Cauchy's contribution to mathematics. His works opened a new era in mathematics, “...the so-called “arithmetization” of all mathematics begins.” Thanks to Cauchy's work, mathematical analysis firmly and deservedly occupied one of the main places in mathematics. Cauchy's methods became widespread and were used and perfected throughout the 19th century. Cauchy's ideas and methods are fruitfully used and generalized by modern mathematicians today.

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This chapter studies the operation of passage to the limit - the main operation of mathematical analysis. First, let's consider the limit of a function of a natural argument, since all the main results of the theory of limits are clearly visible in this simple situation. Next, consider the limit at a point of a function of a real variable.

2.1 Sequence limit

2.1.1 Definition and examples

Definition 2.1. A function f: N → X whose domain is the set of natural numbers is called a sequence.

The values of f(n), n N, are called the terms of the sequence. They are usually denoted by the symbol of the element of the set into which the mapping occurs, providing the symbol with the corresponding index (an argument to the function f): xn = f(n). The element xn is called the nth member of the sequence. In this regard, the sequence is often denoted by the symbol (xn) or (xn)+ n=1 ∞, and is also written in the form x1, x2,. . . , xn , . . . .

Later in this chapter we will consider only the sequence f: N → R of real numbers.

Definition 2.2. The interval containing the point R is called the neighborhood of this point. The interval (a − δ, a + δ), δ > 0, is called the δ -neighborhood of point a and is denoted U a (δ) or V a (δ) (often written in short: U a or V a ).

Definition 2.3. A number a R is called the limit of a numerical sequence (x n ) if for any neighborhood of a point a there is a number N N such that all elements x n of the sequence whose numbers are greater than N are contained in U a . At the same time they write

n lim→∞ xn = aor lim xn = aor xn → aas n → ∞.

In logical symbolism, definition 2.3 looks like:

a R. a = lim xn Ua N = N(Ua ) N: n > N xn Ua .

Since Ua (ε) = (a − ε, a + ε) = (x R: |x − a|< ε}, то часто употребляют следующую равносильную формулировку определения2.3

Definition 2.4. A numbera is called the limit of a number sequence(x n) if for any positive number ε there is a numberN = N(ε) such that all members of the sequence with numbersn > N satisfy the inequality|x n − a|< ε .

Accordingly, in logical symbolism this definition has the form: a R, a = lim xn ε > 0 N = N(ε) N: n > N |xn − a|< ε

Comment. The first terms of the sequence do not affect the existence and magnitude of the limit if it exists.

The following geometric interpretation of Definition 2.3 of the limit of a sequence is sometimes useful:

A number a is called the limit of the sequence(x n) if outside any neighborhood of the point a there are no more than a finite number of members of the sequence(x n).

It is clear that if outside some neighborhood of point a there is infinite number terms (xn), then a is not a limit of (xn).

Let's look at a few examples.

Example 2.1. If (xn) : xn = c, then lim xn = c, since all terms of the sequence, starting from the first, belong to any neighborhood

Example 2.2. Let us show that the sequence (xn) : xn =

has a limit and lim xn = 0.

Let us fix ε > 0. Since

≤ n

< ε для n >

Then, assuming N = max(1, ), we get:

|xn | ≤

Therefore, ε > 0 N = max(1, ) N: n > N |xn |< ε.

Comment. At the same time we proved that lim

Example 2.3. Let us show that lim

0 if q > 1.

Since q > 1, then q = 1 + α, where α > 0. Therefore n > 1 by Newton’s binomial formula

qn = 1 + nα +n(n − 1) α2 + · · · + αn > nα.



It follows that							N > 1. Let us fix ε > 0 and set

N = max(1, ) and we get that

So, ε > 0 N = max(1, ) N: n > N |1/qn |< ε.

Example 2.4. Let us show that the sequence (xn) : xn = (−1)n has no limit.

For any number a, we indicate a neighborhood outside of which there is an infinite set of terms of the given sequence. To do this, we fix a point a R and consider its unit neighborhood Ua (1) = (a − 1, a + 1). Since x2k = 1, x2k+1 = −1, k N, and at least one of the numbers +1 or −1 does not belong to Ua (1), then outside Ua (1) there is an infinite number of terms of the sequence (xn). Therefore, the number a is not its limit. Due to the arbitrariness of the number a, we conclude that @ lim xn.

Definition 2.5. A number sequence whose limit is a number is called convergent. All other sequences are called divergent.

In logical symbolism, Definition 2.5 has the form: (xn) converges a R: lim xn = a.

divergent, and the sequence ((−1)n ) is divergent.

2.1.2 Properties of convergent sequences

Theorem 2.1. A sequence cannot have two different limits.

Let the number sequence (xn) have two different limits a and b. For definiteness, we will assume that a< b. Положим

ε = b − 2 a. By Definition 2.4 of the limit of a sequence we find N1 and

n −

such that

n > N, that is

| n −

Then n > N = max(N1 , N2 )

< xn <

Which can't happen.

Definition 2.6. Number sequence(xn) is called bounded above (respectively, below or bounded) if the set X = (x n | n N) is bounded from above (from below or bounded). If X is an unbounded set, then(xn) called an unbounded sequence.

Taking into account Definitions 2.1 and 2.2 we have:

(xn ) bounded above M R: n N xn ≤ M, (xn ) bounded below M R: n N xn ≥ M, (xn ) bounded M > 0: n N |xn | ≤ M,

(xn ) is not limited M > 0 n N: |xn | > M.

Theorem 2.2. The convergent sequence is limited.

Let the sequence (xn) converge and lim xn = d. Assuming ε = 1 in Definition 2.4, we find a number N such that |xn − d|< 1, n >N, that is, d − 1< xn < d + 1, n >N. Let us introduce the following notation:

a = min(x1 , x2 , . . . , xN , d − 1), b = max(x1 , x2 , . . . , xN , d + 1).

Then a ≤ xn ≤ b, n N.

Comment. Boundedness of the sequence is a necessary but not sufficient condition for convergence (see example 4) .

Theorem 2.3. If the number sequence(x n) converges and lim x n = a , then the sequence(|x n |) converges and lim |x n | = |a|.

Since a = lim xn , then ε > 0 N = N(ε) N: n > N |xn − a|< ε.

It follows that n > N ||xn | − |a|| ≤ |xn − a|< ε.

Remark 1. From Theorem 2.3 and Example 3 it follows that for |q| > 1

lim q n = 0.

Remark 2. The converse of Theorem 2.3 does not hold.