limit

Concept of calculus

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"Limit" is a branch of mathematics - the basic concept of calculus. In a broad sense, "limit" means "infinitely close and never reach". In mathematics, "limit" refers to a variable in a function that gradually approaches a certain value A and "can never coincide with A" ("can never be equal to A, but taking equal to A is enough to obtain high-precision calculation results) The change of this variable is artificially defined as "always close without stopping", and it has a "trend of constantly getting very close to point A.". Limit is a description of "changing state". This variable is always Approach The value A of is called“ Limit value " (Of course, it can also be represented by other symbols).

The above is a popular description of the connotation of "limit", and the strict concept of "limit" is ultimately Cauchy and Weierstrass And others.

Chinese name: limit
Foreign name: limit
Discipline: Calculus
application area: Calculus

representative figure: Cauchy and Weierstrass
Limit number: lim
Field: mathematics
Type: Concept of calculus

catalog

one brief introduction
two Production and development
▪ The thinking function of limit thought
▪ Established concept
▪ Limit thought of solving problems

brief introduction

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The idea of limit is an important idea of modern mathematics. Mathematical analysis is based on the concept of limit Limit theory (including series) as the main tool to study the function of a subject.

The idea of limit refers to the concept of "using limit to analyze and solve problems Mathematical thinking " 。

use Limit thought The general steps to solve the problem can be summarized as follows:

For the unknown quantity to be investigated, first try to correctly conceive another variable related to its change, and confirm that the 'influence' trend result of this variable through the infinite change process is very precise, which is approximately equal to the unknown quantity sought; Using the limit principle, the results of the unknown quantity under investigation can be calculated.

The limit thought is Calculus The basic idea of mathematical analysis A series of important concepts in, such as continuity of function, derivative (getting maximum or minimum for 0) and definite integral And so on are defined by means of limits. If you want to ask: "What is mathematical analysis?", you can generally say: "Mathematical analysis is a subject that studies functions with limit thought, and the error of calculation results is too small to imagine, so it can be ignored.

Production and development

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(1) Origin

Like all scientific methods of thinking, limit thinking is also the product of abstract thinking in the brain of social practice. The idea of limit can be traced back to ancient times, such as the motherland Liu Hui Of Cyclotomy It is the application of an original and reliable limit thought of "approaching constantly" based on the research of intuitive graphics; ancient Greek Human Exhaustion method It also contains the idea of limit, but because of the "fear of 'infinity'" of the Greeks, they avoid obviously artificially "taking the limit", but rely on indirect evidence—— Reduction to absurdity To complete the relevant certification.

In the 16th century, Dutch mathematicians Steven On inspection Triangle center of gravity In the process of, he improved the exhaustion method of the ancient Greeks. With the help of geometric intuition, he boldly used the limit thought to think about problems, and gave up the proof of the method of returning to Miao. In this way, he inadvertently "pointed out the direction of developing the limit method into a practical concept".

(2) Development

The further development of limit thought is closely related to the establishment of calculus. Europe in the 16th century The rudiments of capitalism During this period, the productivity was greatly developed, and a lot of problems were encountered in production and technology Elementary mathematics The method of "can not be solved, and requires mathematical breakthrough" only research Constant‘ It is the social background to promote the development of 'limit' thinking and establish calculus to find new tools that can describe and study the movement and change process.

at first Newton And Leibniz Infinitesimal They built calculus on the basis of concepts, and later, because of logic difficulties, they all accepted the idea of limit to varying degrees in their later years. Newton uses the ratio of the distance change Δ S' to the time change Δ t '“

”Representing a moving object average velocity , let Δ t infinitely approach to zero, get the instantaneous velocity of the object, and then lead to the concept of derivative and the theory of differential calculus. He realized the importance of the concept of limit, and tried to use the concept of limit as the basis of calculus. He said: "If the ratio of two quantities to quantity keeps getting equal in a finite time, and they are close to each other before the end of this time, so that the difference is less than any given difference, then they will eventually become equal". But Newton's concept of limit was also based on geometric intuition, so he could not get a strict expression of limit. The limit concept used by Newton is only close to the following intuitive language description: "If n increases infinitely, a n It is infinitely close to the constant A, so a n Take A as the limit.

It was precisely because of the lack of strict limit definition at that time that calculus theory was doubted and attacked by people for scientific theory. For example, in physics, whether Δ t (change) is equal to zero? If it is said to be zero, (because truth will become wrong if its scope of application is expanded infinitely): how can it be used division What about? (Actually, the change cannot be 0). But people think that if it is not zero, how can computers and functions get rid of those "tiny quantities" that contain it when they are deformed? At that time, people didn't understand and wanted to calculate variables without any error, which led to people thinking that there was a paradox History of mathematics The reason for the infinitesimal paradox mentioned above. British philosopher and archbishop Berkeley The attack on calculus was the most intense. He said that the derivation of calculus was "clear sophistry". The history and success of scientific development show that his view is wrong.

Berkeley's fierce attack on calculus, on the one hand, was to serve religion; on the other hand, because calculus at that time lacked a solid theoretical basis and flexible solutions, even the famous Newton could not get rid of the confusion in the 'limit concept'. This fact shows that to understand the concept of "limit", it is a dynamic process of infinite change of quantity. Of course, the trend direction of small variables can be very precisely approximated to a certain constant. This is the ideological basis for establishing a strict calculus theory, and has great significance as a tool for scientific research in epistemology.

(3) Perfect

The perfection of limit thought is closely related to the strictness of calculus. For a long time, many people have tried to solve the problems of the theoretical basis of calculus with "complete satisfaction", but they failed to achieve their wishes. This is because the research object of mathematics has expanded from constant to variable, and people are accustomed to thinking and analyzing problems with constant. The understanding of the unique concept of "variable" is not very clear; The difference and relationship between "variable mathematics" and "constant mathematics" are still poorly understood; The unity of opposites between "finite" and "infinite" is still unclear. In this way, people can not adapt to the new development of 'variable mathematics' because of the rigid thinking of the traditional way of thinking used to deal with constant mathematics. The ancient people's habit of using the old concept constant cannot explain the dialectical relationship between the scientific conclusion that "zero" and "non zero value that is infinitely close to zero" can be artificially small distance jump to equal mutual transformation.

In the 18th century, Robbins, d'Alembert and Royrier clearly stated that limit must be taken as the basic concept of calculus, and they all made their own definitions of limit. Among them, d'Alembert's definition is: "one quantity is the limit of another quantity, if the second quantity is closer to the first quantity than any given value", the connotation of its description is close to the correct definition of limit; However, these definitions cannot get rid of the dependence on geometric intuition. This is the only way of thinking, because most of the concepts of arithmetic and geometry before the 19th century were based on Geometric quantity Conceptual. In fact, "visualization" is not a synonym for backward thinking, and the study of geometric intuition is not a synonym for backward thinking, because today it is still possible to use function "mapping" as a graph to study more complex trend problems. If there is a trend, the concept of limit can be established. For example, "visualized" graphics can more intuitively prove that a proposition that has no rule to describe and can't attack users for a long time cannot be established; (or another function can be established), and then make specific mathematical proofs of "symbolic way" respectively.

First of all, what gives the correct definition of 'derivative' with the concept of limit is Czech Republic Mathematician Boer Chano defined the derivative of function f (x) as Quotient difference

He emphasized that f '(x) is not the quotient of two zeros. Polzano's thought is valuable, but he has not yet described the 'essence of limit' clearly.

In the 19th century, French mathematicians Cauchy On the basis of previous work, he elaborated the "limit concept" and its theory in a more complete way. He pointed out in the "Analysis Course" that "when the value of a variable taken one after another is infinitely close to a fixed value, the difference between the value of the variable and the fixed value will eventually be as small as possible, and this fixed value is called the limit value of all other values, especially when the value (absolute value) of a variable is If it is infinitely reduced so that it converges to the limit 0, it is said that this variable becomes infinitesimal. "

Cauchy regards the infinitesimal as a "variable with 0 as the limit", which correctly establishes the concept of "infinitesimal" as a way of "treating it as if it is zero but not zero", that is, in the process of variable change, its value is actually not equal to zero, but its trend of change is toward "zero", which can be infinitely close to zero. Then people can use "equal to 0" to deal with it, which will not produce wrong results.

Cauchy tried to eliminate the geometric intuition in the concept of limit, (but "geometric intuition" is not a negative thing, and we can also use our imagination when studying functions - "the variable image of dynamic trend, assuming that after being magnified to a huge astronomical multiple, we will never see that the variable value 'coincides with 0', so it will be more" clear "to use inequality) Make a clear definition of limit, and then complete Newton's wish. However, there are still descriptive words in Cauchy's narration, such as "unlimited approach" and "as small as possible", which are relatively easy to understand. It is easier to understand the concept, so its definition still retains the intuitive traces of geometry and physics, which can be divided into two parts. It is also beneficial to have more intuitive traces, But it is easier to understand the concept of 'limit' by combining the following abstract definitions.

In order to eliminate the intuitive traces in the concept of limit, Weierstrass The static abstract definition of limit is proposed, which provides a strict theoretical basis for calculus. The so-called x n → x means: "If for any ε>0, there is always a natural number N, so that when n>N, the inequality | x n -X |<ε is always true ".

This definition, with the help of inequality, quantitatively and concretely depicts the relationship between the two "infinite processes" through the relationship between ε and N. Therefore, such a definition should be a relatively strict one at present, which can be used as the basis of scientific demonstration and is still used in mathematical analysis books. In this definition, only 'number and its size relationship' is involved. In addition, only given, existing, any and other words are used. The word "approaching" has been cast off, and the intuition of motion is no longer resorted to. (However, the concept of 'limit' cannot be understood without abandoning 'movement trend', otherwise it is easy to lead to 'unscientific introduction of constant concept into the field of calculus')

Constants can be understood as' unchanging quantities'. Before the advent of calculus, people used to use static images to study mathematical objects. Since the advent of analytic geometry and calculus, the thinking mode of motion considering 'variation' has entered the field of mathematics, and people have mathematical tools to conduct dynamic research on the change process of physical quantities and other things. Later, Weierstrass, who established ε - N language, used static definitions to describe the change trend of variables. This spiral evolution of "static dynamic static" reflects the dialectical law of mathematical development.

The thinking function of limit thought

The limit thought is widely used in modern mathematics and even physics, which is determined by its inherent thinking function. The limit thought reveals the unity of opposites between variable and constant, infinity and finity, which is materialistic dialectics Law of unity of opposites Application in the field of mathematics. With the help of limit thought, people can know infinity from finiteness, change from invariance, curve from straight line to form, qualitative change from quantitative change, and accuracy from approximation.

The concept of "infinity" is different from that of "finity" in essence, but they are related. "infinity" is the concept of abstract thinking in the brain, which exists in the brain. "Limited" is the mapping of the "quantity" of the ever-changing things in the objective reality. The "infinite" in line with the objective reality law belongs to the whole. According to the axiom, the whole is greater than the local thinking.

"Change" and "invariance" reflect the two different states of things: movement and change, and relative stillness, but they can transform each other under certain conditions. This transformation is "one of the powerful levers in mathematical science". For example, physics Variable speed linear motion Of Instantaneous velocity , using primary There is no way to solve it, but the difficulty lies in Variable speed linear motion Of Instantaneous velocity Is a variable, not a constant. For this reason, people first use the "uniform speed" calculation method instead of the "variable speed" calculation in a small time interval to find the average velocity The instantaneous speed in a small time is defined as seeking the "speed limit", which is the precise result of seeking the "limit" of "changing at a certain time" from the "unchanged" form with the aid of the limit thinking method.

There are essential differences between curved and linear images, but they can also be transformed into each other under certain conditions, such as Engels Said: "The straight line and curve differential China has finally become equal ". Making good use of the unity of opposites is one of the important means to deal with mathematical problems. The area of a figure formed by straight lines is easy to calculate; But the area of a graph formed by a curve cannot be accurately solved by elementary mathematics. the ancients Liu Hui Use "" circle inscribed polygon to approximate circle area "; People use "area transformed into rectangle" to approximate Curved trapezoid The area of, and so on, are all based on the thinking method of the limit, starting from the straight line to understand the solution of the curve problem.

The idea of infinite approximation to the "true value" (the conclusion has no error at all) Mathematical research It plays an important role in the work. For example, inscribed on any circle Regular polygon When the number of sides is doubled, the approximate answer to the circle area is Inscribed regular polygon Area of. People are constantly doubling the number of sides. After the infinite process, the polygon will "change" into a "false circle" that is not much different from the real circle area. Each step of "the change in the number of sides" can be accumulated using the original "constant formula" to get the "circle area" that is closer and closer to the real value, and more and more new small triangle bottom edges on the edge of the circle, The limit of the sum of its long sides is equal to the product of "half of the circumference" and the radius to calculate the circle area (that is, the application of the limit concept). The trend limit is getting closer and closer Circular area 。 This is to solve the problem of circle area by means of limit thinking method, which is the same as other problems.

When solving problems with the concept of limit, first use the traditional thinking, use the constant thinking of 'low mathematical thinking' to establish a certain function (calculation formula), then think of a way to deform the total area of the image unchanged, and then calculate the limit of a corresponding variable, which can solve the problem. Finding the limit value in the transformation of "identity" is an important trick of applying mathematics to the calculation of real variables. The "partial sum"“ average velocity ”, "area method of inscribed regular polygon", which are the corresponding "approaching value of infinite series"“ Instantaneous velocity ”The most accurate approximate value of "circle area" can be obtained by using the limit thought. This is all based on the thinking method of limit. People can also achieve precise calculation results by 'infinite approximation'. With this new method - the limit thinking of calculus, the problem of 'directly calculating functions with variable quantities by constant method but no ready-made formula is available, so the calculation result error is large' can be satisfactorily solved.

Established concept

The thinking method of limit runs through mathematical analysis Always in the course. It can be said that almost all concepts in mathematical analysis are inseparable from limits. In almost all works of mathematical analysis, the method of thinking of function theory and limit is introduced first, and then the method of thinking of limit is used to give continuous function , derivative definite integral , convergence and divergence of series Multivariate function Of partial derivative ， Generalized integral Convergence and divergence, multiple integral sum of curvilinear integral And Surface integral The concept of. For example:

(1) The definition of continuous function at a point is when independent variable When the increment of tends to zero, function value The increment of tends to the zero limit.

(2) The definition of the derivative of a function at a point is the ratio of the increment of the function value to the increment of the independent variable, and the limit at that time.

(3) The definite integral of a function at a point is defined as the limit of the sum of integrals when the fineness of the division tends to zero.

（4） Number series The convergence and divergence of is defined by the limit of partial sum sequence.

（5） Generalized integral yes definite integral Where is the limit of any real number greater than, etc.

Limit thought of solving problems

’The method of limit thinking is mathematical analysis An essential and important method of even all higher mathematics is also the development of 'mathematical analysis' and further thinking on the basis of' elementary mathematics'. The reason why mathematical analysis can solve many problems that cannot be solved by elementary mathematics (such as calculating instantaneous speed, curve arc length, curved edge area, volume of curved surface body, etc.

By examining the trend and tendency of a series of innumerable more and more precise approximations of certain functions, people can scientifically convert the extreme value of that quantity Exact value To determine, this requires the use of the concept of limit and the above limit thinking methods. It should be believed that it is scientific to use the limit thinking method, because extremely accurate conclusions can be obtained through the limit function calculation method.

sequence limit

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definition

Can define a sequence {x n }Convergence of:

Set {x n }Is a set of infinite real number sequences. If there is a real number a, for any positive number ε (no matter how small), ∃ N>0, so that the inequality | x n -A |<ε is always true on n ∈ (N,+∞), then the constant a is said to be series {x n }Limit of, or number sequence {x n } convergence On a. record as

。

If the above conditions are not true, that is, there is a positive number ε, no matter how many positive integers N are, there is a certain n>N, so that | x n -A | ≥ ε, that is, the sequence {x n }Does not converge to a. If {x n }Not converging to any constant, it is called {x n }Divergence.^[1-2]

Understanding of definitions:

1、 Arbitrariness of ε The function of ε in the definition is to measure the general term of sequence

The closeness to the constant a. The smaller ε is, the closer it is; The positive number ε can be arbitrarily reduced, indicating that x n And the constant a can be close to any degree of continuous close. However, although ε has its arbitrariness, once it is given, it is temporarily determined so that it can be used to find N by functional laws;

And because ε is an arbitrarily small positive number, ε/2, 3 ε, ε two They are also in any small range of positive numbers, so their values can be approximated to replace ε. At the same time, just because ε is an arbitrarily small positive number, we can limit ε to be less than a certain positive number.

2、 Correspondence of N In general, N increases with the decrease of ε, so N is often written as N (ε) to emphasize the dependence of N on the change of ε. But this does not mean that N is uniquely determined by ε: (for example, if n>N makes | x n -A |<ε is true, then obviously n>N+1, n>2N, etc. also make | x n -A |<ε holds). What matters is the existence of N, not the size of its value.

3. From Geometric meaning Look up, "When n>N, there is an inequality | x n -A |<ε is true "means that all subscripts greater than N

All fall within (a - ε, a+ε); Beyond (a - ε, a+ε), the sequence {x n }There are at most N (limited) items in. In other words, if there is some ε zero >0, make the number sequence {x n }There are infinitely many terms in (a - ε zero ，a+ε zero ）Otherwise, {x n }It must not take a as the limit.

Note the geometric meaning: 1. There are at most N (finite) points outside the interval (a - ε, a+ε); 2. All other points x N+1 ,x N+2 ,... (Unlimited) all fall within the neighborhood. These two conditions are indispensable. If a sequence can meet these two requirements, then the sequence converges to a; If a sequence converges to a, both conditions can be satisfied. In other words, if you only know that there is {x in the interval (a - ε, a+ε) n }It can not be guaranteed that there are only finite terms except (a - ε, a+ε), and it is impossible to get {x n }If it converges to a, pay special attention to this point when doing judgment questions.

nature

1、 Uniqueness : If the limit of the sequence exists, the limit value is unique, and any of its Child column The limit of is equal to the original sequence.

2、 Boundedness : If a sequence is convergent (has a limit), it must be bounded.

However, if a series Bounded, this sequence may not converge. For example: "1, - 1, 1, - 1,..., (- 1) n+1 ”

3、 Number assurance : If

(or<0), then for any m ∈ (0, a) (when a<0 is m ∈ (a, 0)), there is N>0, so that when n>N there is

(corresponding x n <m）。

4、 Inequality preserving : Set number sequence {x n }And {y n }Uniform convergence. If there is a positive number N, there is x when n>N n ≥y n , then

(If the condition is changed to x n >y n , the conclusion remains unchanged).

5、 Compatibility with real number operations : For example, if two sequences {x n } ，{y n }All converge, then the sequence {x n +y n }It also converges, and its limit is equal to {x n }The limit sum of {y n }The sum of the limits of.

6、 Relationship with child columns : Number sequence {x n }And any ordinary Child column Both are convergent or divergent, and have the same limit at the time of convergence; Sequence {x n }The sufficient and necessary condition for convergence is: sequence {x n }Any nontrivial subsequences of are convergent.

Monotone convergence theorem

Monotone bounded sequence must convergence 。^[3]

Cauchy convergence principle

Set {x n }Is a sequence of numbers. If there is N ∈ Z * for any ε>0, as long as n satisfies n>N, there is | x for any positive integer p n+p -x n |<ε, such sequence {x n }Is called Cauchy sequence 。

This asymptotic stability is equivalent to convergence. mean Sufficient and necessary conditions 。

topological space

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Set X to topological space ，

Is in X sequence 。 Then a point a ∈ X is called sequential limit , if any to a Neighborhood U， Existence of n zero , meet the requirements for any n ≥ n zero ，x n ∈U。^[4]

Function limit

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Limit of function when independent variable approaches finite value

definition: Let function f (x) be at point x zero One of Decanter neighborhood If there is a constant a, for any given positive number ε

, make inequality

stay

Time constant is established, then the constant

It's called a function

When

The limit of time, recorded as

。^[1-2]

If the function

When

When a is not taken as the limit, there is a positive number ε. For any positive number δ, when

When,

。

(Explanation: when

Hour

Convergence to

, we must be able to prove that x is close enough to x zero When,

And limits

Gap of Less than any small specified error. And when

Hour

Not converging to

We can prove that whether x or x zero The distance between f (x) and a cannot be less than a specified error.)

Limit of function when independent variable approaches infinity

definition: Let function f (x) be defined when | x | is greater than a positive number. If there is a constant a, for any given positive number ε, there is always a positive number M, so that when x satisfies the inequality