微积分 | 极限思想的发展及其应用

注：本文为 “微积分 | 极限思想” 相关合辑。
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中文引文，略作重排。
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Development of Extreme Thoughts in Calculus and Its Application in Mathematical Analysis

微积分中极限思想的发展及其在数学分析中的应用

Dongchun Tang

School of science, Tibet University, Tibet, Lhasa, 850000

*Corresponding author e-mail: tangmiaoji@sina.cn

Abstract

摘要

Extreme thought is the basic idea of calculus. A series of important concepts in mathematical analysis, such as the continuity of functions, derivatives and definite integrals, are all defined with the help of extreme thought. The generation and improvement of extreme thought is under the need of society and it has created a new impetus for the development of mathematics, and has become the basis and starting point of modern mathematical thought and methods. In order to deepen the public’s understanding of extreme thought, this article has studied the development and application of extreme thought. First of all, the concept and emergence of extreme thought are briefly introduced, and then the development process of extreme thought is analyzed from three stages: the germination period, the development period, and the perfect period, and finally the application of extreme thought in mathematical analysis is explained from three aspects.
极限思想是微积分的基本思想。数学分析中的一系列重要概念，如函数的连续性、导数、定积分等，均借助极限思想来定义。极限思想的产生与完善源于社会需求，为数学发展注入了新动力，已成为现代数学思想与方法的基础和起点。为加深大众对极限思想的理解，本文对极限思想的发展与应用展开研究。首先简要介绍极限思想的概念及产生，随后从萌芽期、发展期、完善期三个阶段分析极限思想的发展历程，最后从三个方面阐述极限思想在数学分析中的应用。

1. Introduction

1. 引言

As a kind of philosophical and mathematical thought, extreme thought has sprouted from ancient thoughts to the present complete limit theory. Its long and tortuous course of evolution is filled with the hard work, wisdom, rigorous, earnest, and diligent struggle of many philosophers and mathematicians [1]. The evolution of extreme thoughts is a side reaction of the entire process of human understanding of the world and the transformation of the world for thousands of years. The application of extreme ideas is ubiquitous. Understanding and grasping the rational application of extreme thought allows us to quickly find solutions to problems in the process of solving practical problems and improve practical results [2].
极限思想作为一种兼具哲学性与数学性的思想，从古代思想萌芽发展至如今完备的极限理论。其漫长而曲折的演变历程，凝聚了众多哲学家与数学家的心血、智慧，以及严谨认真、不懈奋斗的精神 [1]。数千年来，极限思想的演变是人类认识世界、改造世界全过程的一种映射。极限思想的应用无处不在，理解并掌握极限思想的合理运用，能让我们在解决实际问题时快速找到解题思路，提升实践效果 [2]。

2. Emergence and Development of Extreme Thoughts

2. 极限思想的产生与发展

2.1 The Concept Of Extreme Thinking

2.1 极限思想的概念

Limits are infinitely approaching a fixed value. Limits can be divided into series limits and function limits [3].
极限是无限趋近于某一固定值的过程。极限可分为数列极限与函数极限 [3]。

Definition 1: Let { a n } \{a_{n}\} {an​} be a sequence of numbers, $a$ is a fixed number. For any given positive number $\epsilon$ , there is always a positive integer $N$ , so that when $n>N$ , there is $|a_n-a|<\epsilon$ . Then the sequence { a n } \{a_{n}\} {an​} converge to $a$ , the fixed number $a$ is called a limit of sequence { a n } \{a_{n}\} {an​} , and recorded as ${\lim }_{n\to \infty }{a}_n=a$ , or $a_n\to a(n\to \infty )$ .
定义 1：设 { a n } \{a_{n}\} {an​} 为一数列， $a$ 为一固定常数。对于任意给定的正数 $\epsilon$ ，总存在正整数 $N$ ，使得当 $n>N$ 时，有 $|a_n-a|<\epsilon$ 成立，则称数列 { a n } \{a_{n}\} {an​} 收敛于 $a$ ，固定常数 $a$ 称为数列 { a n } \{a_{n}\} {an​} 的极限，记为 ${\lim }_{n\to \infty }{a}_n=a$ ，或 $a_n\to a(n\to \infty )$ 。

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author (s) and the title of the work, journal citation and DOI.
本作品内容可依据知识共享署名 3.0 许可协议（Creative Commons Attribution 3.0 licence）使用。对本作品的任何进一步传播，都必须保留对作者、作品标题、期刊引用信息及 DOI 的标注。

Published under licence by IOP Publishing Ltd
2020 International Conference on Applied Physics and Computing (ICAPC 2020) Journal of Physics: Conference Series
IOP Publishing 1650 (2020) 032204
doi:10.1088/1742-6596/1650/3/032204

Definition 2: Let function $f(x)$ be defined in a hollow neighborhood of point $x_0$ . $A$ is a fixed number, if for any $\epsilon >0$ , there is a positive number $\delta$ , so that when $0<|x-x_0|<\delta$ , there is $|f(x)-A|<\epsilon$ . Then $A$ is the limit of the function $f(x)$ as $x$ tends to $x_0$ , recorded as ${\lim }_{x\to x_0}f(x)=A$ .
定义 2：设函数 $f(x)$ 在点 $x_0$ 的某一空心邻域内有定义， $A$ 为一固定常数。若对于任意 $\epsilon >0$ ，总存在正数 $\delta$ ，使得当 $0<|x-x_0|<\delta$ 时，有 $|f(x)-A|<\epsilon$ 成立，则称当 $x$ 趋近于 $x_0$ 时，函数 $f(x)$ 的极限为 $A$ ，记为 ${\lim }_{x\to x_0}f(x)=A$ 。

2.2 Emergence of Extreme Thoughts

2.2 极限思想的产生

Like all scientific thinking methods, extreme thoughts are also a product of social practice. Extreme thoughts can be traced back to ancient times. Dutch mathematician Steven improved the exhaustion method of the ancient Greeks in the process of examining the center of the triangle. He used extreme thoughts to think about the problem and gave up the proof of the fallacy method. Therefore, he inadvertently “proposed to develop the limit method into a method using concepts” [4].
与所有科学思维方法相同，极限思想也是社会实践的产物，其源头可追溯至古代。荷兰数学家史蒂文（Steven）在研究三角形重心问题时，对古希腊人的穷竭法进行了改进。他运用极限思想思考问题，摒弃了归谬法的证明思路，从而无意间 “提出将极限方法发展为一种运用概念的方法”[4]。

However, due to the limitations of historical conditions in the early days of calculus, people’s understanding of his basic concepts and their relationships could not break through the limitations of mechanics and geometrical intuition. Many concepts did not have a precise mathematical definition. The derivation of the formula is still in a chaotic logic.
然而，在微积分发展初期，受历史条件限制，人们对其基本概念及概念间关系的理解未能突破力学与几何直观的局限。许多概念缺乏精确的数学定义，公式的推导也存在逻辑混乱的问题。

2.3 Development of Extreme Thoughts

2.3 极限思想的发展

2.3.1 The budding period of extreme thoughts

2.3.1 极限思想的萌芽期

Far more than 2000 years ago, in the understanding of infinity, people’s extreme thoughts and methods were inevitably nurtured in them. In China, the famous book “Zhuangzi・Tianxia Pian” states: “One hammer, take half of the day, endlessly. This can reflect the early limit of material in China. There is a profound understanding of separability and continuity. Although these understandings belong to philosophy, they have reflected the germination of extreme thoughts [5]. Liu Hui used the extreme thought to deal with the problem many times in the commentary"Nine Chapters of Arithmetic”, and he was more skilled in using it, indicating that he had a very deep understanding of the extreme thought at that time. He already had the concept and method of the limit application on the basis of intuition. It is based on the theory of"circumcision"that Liu Hui obtained the emblem rate. By the fifth century AD, the great mathematician and scientist Zu Chongzhi (429-500) of the Southern and Northern Dynasties’“Stitching"in the same way, using the"circumcision” method to calculate the 24576 polygons: $3.1415926<\pi <3.1415927$ . Zu Chongzhi’s achievements are nearly 1,000 years ahead of the world.
两千多年前，在对 “无穷” 的认识过程中，极限思想与方法便已开始孕育。在中国，名著《庄子・天下篇》中有 “一尺之棰，日取其半，万世不竭” 的论述，这体现了中国古代对物质可分性与连续性的深刻认识。尽管这些认识属于哲学范畴，但已反映出极限思想的萌芽 [5]。刘徽在《九章算术注》中，多次运用极限思想解决问题，且运用娴熟，可见他在当时对极限思想已有较深刻的理解，在直观基础上具备了运用极限的思想与方法。刘徽正是基于 “割圆术” 理论得到了徽率。到了公元 5 世纪，南北朝时期伟大的数学家、科学家祖冲之（429-500）沿用 “割圆术”，通过计算正 24576 边形的边长，得出 $3.1415926<\pi <3.1415927$ 。祖冲之的这一成果比世界同类成果早了近 1000 年。

In foreign countries, when studying the problem of drawing a circle, Antifen thought of using an inscribed regular polygon with an increasing number of sides to approach the area of the circle. When the number of sides of the polygon continued to double, the gap between the inscribed regular polygon and the circle was gradually “exhausted”, and Bryson (ca. 450 BC) proposed from the opposite direction to approach the circle by the area of the circumscribed regular polygon. In the 4th century BC, the ancient Greek mathematician Eudoxus created a stricter general method for determining area and volume—the “exhaustion method”. This method assumes the infinite separability of quantities, and the following propositions basis: “If you subtract a part that is not less than half of it from any amount, and subtract another part that is not less than half of it from the rest, and continue, you will end up with a less than any given same amount of quantity”. Applying the exhaustion method, Eudoxus (about 400-347 BC) correctly proved that “the area of a circle is proportional to the square of the diameter” and “the volume of the ball is proportional to the cube of the diameter”. His exhaustion method has also reflected the idea of limit theory. After Eudoxus, Archimedes used the exhaustion method to find the area of a series of geometric figures. He approached the perimeter with a sufficient number of “inscribed” and “external” fans. The plane figure formed, which is very different from China’s “circumcision” theory, is essentially an extreme idea. Archimedes (287-212 BC) was born in Syracuse (now Sicily, Italy)). He is exceptionally intelligent and accomplished, and is known as the greatest mathematician and scientist in ancient times. His famous masterpieces include “Measurement of Circles”, “On Spheres and Cylinders”, “On Split Cone Surfaces and Spheres”, “Parabolic Bow Quadrature”, “On Spiral”, “Grit Calculation”, etc. He cleverly combined the method of exhaustion of Eudoxus with the atomic view of Democritus through rigorous calculations, solved a large number of calculation problems. He broke through the traditional finite operation, adopted the idea of infinite approximation, and divided the amount of required product into many tiny units, and then used another groups of small units that are easy to calculate sums for comparison. His concept of infinitesimals was used by Newton as the basis of calculus until the 17th century. Archimedes’ outstanding achievements enriched the content of ancient mathematics, and the depth of his thoughts and the rigor of his discourse were extremely rare at the time, so he was called “The God of Mathematics” and the “Four Masters of Mathematics” together with Gauss, Euler and Newton before the 19th century.
在国外，安提丰（Antifen）在研究化圆为方问题时，提出用边数不断增加的圆内接正多边形去逼近圆的面积。当多边形的边数不断加倍时，圆内接正多边形与圆之间的差距逐渐 “穷竭”；而布里松（Bryson，约公元前 450 年）则从相反方向提出，用圆外切正多边形的面积去逼近圆的面积。公元前 4 世纪，古希腊数学家欧多克索斯（Eudoxus）创立了一种更严格的确定面积与体积的一般方法 ——“穷竭法”。该方法假设量具有无限可分性，并以如下命题为基础：“从任意量中减去不小于其一半的部分，再从余下部分中减去不小于其一半的部分，如此继续下去，最终会得到一个小于任意给定同类量的量”。欧多克索斯（约公元前 400-347 年）运用穷竭法，正确证明了 “圆的面积与直径的平方成正比” 以及 “球的体积与直径的立方成正比”，其穷竭法中已蕴含极限理论的思想。欧多克索斯之后，阿基米德（Archimedes）运用穷竭法求得了一系列几何图形的面积。他用足够多的 “内接” 与 “外切” 扇形去逼近周长，所形成的平面图形虽与中国的 “割圆术” 理论存在较大差异，但本质上均为极限思想的体现。阿基米德（公元前 287-212 年）出生于叙拉古（现意大利西西里岛），他天资聪颖、成就卓著，被誉为古代最伟大的数学家与科学家。其著名著作包括《圆的度量》《论球与圆柱》《论劈锥曲面体与球体》《抛物线弓形求积》《论螺线》《沙粒计算》等。他通过严谨的计算，将欧多克索斯的穷竭法与德谟克利特（Democritus）的原子论巧妙结合，解决了大量计算问题。他突破传统有限运算的局限，采用无限逼近的思想，将所需计算的量分割成众多微小单元，再用另一组易于求和的小单元进行比较。他提出的无穷小概念，直到 17 世纪仍被牛顿（Newton）用作微积分的基础。阿基米德的卓越成就丰富了古代数学的内容，其思想深度与论述严谨性在当时极为罕见，因此被称为 “数学之神”，并与 19 世纪前的高斯（Gauss）、欧拉（Euler）、牛顿共同被誉为 “数学四巨匠”。

From this, we can see that at the beginning of the development of extreme thoughts in mathematics, the ancients have created a glorious starting point in the limit field.
由此可见，在数学领域中极限思想发展的初期，古人便在极限领域开创了辉煌的起点。

2.3.2 The development period of extreme thoughts

2.3.2 极限思想的发展期

At the end of the 14th century, Europe began to have the germination of capitalism. By the middle of the 15th century, the dissolution of the feudal system, Europe’s productive forces developed rapidly, and the “Renaissance” era began. The development of productive forces also promoted the development of science and technology, at that time, a lot of new questions were raised in astronomy, physics, geography and other aspects around mechanics as the center, and the exploration of these questions promoted the development of related sciences. For example, the birth zone of Copernicus’ “Heliocentric Theory” is a revolution in natural science; due to the study of celestial mechanics, a group of scientists have emerged, such as Steven, Galileo, Kepler, etc., they have also done a lot of research in mathematics, which has laid the foundation and brought opportunities for the development and application of extreme ideas and methods. After the 16th century, Europe was in the embryonic period of capitalism, and productivity had greatly developed. A large number of variables problems occurred in productivity and science and technology such as curve tangent problems, maximum value problems, speed problems in mechanics, work of force, etc., elementary mathematical methods are becoming more and more powerless for this, and new thoughts and new mathematical method to break through the traditional scope of the study are needed to provide new tools to describe and study movement [6].
14 世纪末，欧洲开始出现资本主义萌芽。到 15 世纪中期，封建制度逐渐瓦解，欧洲生产力快速发展，“文艺复兴” 时代来临。生产力的发展推动了科学技术的进步，当时以力学为中心，天文学、物理学、地理学等领域提出了大量新问题，对这些问题的探索又促进了相关学科的发展。例如，哥白尼（Copernicus）“日心说” 的诞生是自然科学领域的一场革命；天体力学的研究催生了史蒂文、伽利略（Galileo）、开普勒（Kepler）等一批科学家，他们在数学领域也开展了大量研究，为极限思想与方法的发展和应用奠定了基础、创造了机遇。16 世纪以后，欧洲处于资本主义萌芽阶段，生产力大幅提升。在生产力与科技领域出现了大量与变量相关的问题，如曲线的切线问题、最大值问题、力学中的速度问题、力的功的计算问题等，初等数学方法对此愈发无能为力，亟需突破传统研究范畴的新思想与新数学方法，为描述和研究运动提供新工具 [6]。

Many mathematicians have made unremitting efforts to solve the above problems, such as Descartes, Fermat, Barrow, Cavalieri, Wallis, etc., and have achieved certain results, especially Newton and Leibniz founded calculus in their work, they all use extreme ideas and methods from different angles. Although their work relies too much on intuitiveness and lacks a strict logical foundation, their efforts and achievements have laid a solid foundation for the further improvement of extreme ideas.
为解决上述问题，笛卡尔（Descartes）、费马（Fermat）、巴罗（Barrow）、卡瓦列里（Cavalieri）、沃利斯（Wallis）等众多数学家付出了不懈努力，并取得了一定成果。尤其是牛顿和莱布尼茨（Leibniz），他们在研究中创立了微积分，且均从不同角度运用了极限思想与方法。尽管他们的研究成果过多依赖直观性，缺乏严格的逻辑基础，但其努力与成就为极限思想的进一步完善奠定了坚实基础。

2.3.3 The perfect period of extreme thoughts

2.3.3 极限思想的完善期

The improvement of extreme thoughts is closely related to the strictness of calculus. For a long time, many people have tried to solve the problem of calculus theory, but they have not been able to achieve it. This is because the research objects of mathematics have changed from constant extended to variables, and people are not very clear about the specific laws of variable mathematics; they still lack a clear understanding of the differences and connections between variable mathematics and constant mathematics; and they are not clear about the unified relationship of finite and infinite opposition. In this way, people are used to the traditional way of thinking of constant mathematics and cannot adapt to the new needs of variable mathematics. The dialectical relationship between “zero” and “non-zero” mutual transformation cannot be explained by the old concepts alone.
极限思想的完善与微积分的严谨化进程密切相关。长期以来，许多学者试图解决微积分理论的严谨性问题，但均未取得成功。这是因为数学的研究对象从常量扩展到变量后，人们对变量数学的具体规律认识不清；对变量数学与常量数学之间的区别与联系缺乏清晰认知；对有限与无限的对立统一关系也不明确。如此一来，人们习惯了常量数学的传统思维方式，无法适应变量数学的新需求，仅靠旧有概念难以解释 “零” 与 “非零” 相互转化的辩证关系。

In the 18th century, Robbins, D’Alembert, and Rouliere and others have clearly stated that the limit must be used as the basic concept of calculus, and they all have defined limits. Among them, D’Alembert’s definition is: “A quantity is the limit of another quantity, if the second quantity is closer to the first quantity than any given value”, it is close to the correct definition of the limit; however, the definition of these people cannot get rid of geometrical intuition. This can only be the case, because most of the concepts of arithmetic and geometry before the 19th century were based on the concept of geometric quantities.
18 世纪，罗宾斯（Robbins）、达朗贝尔（D’Alembert）、罗利尔（Rouliere）等人明确提出，必须将极限作为微积分的基本概念，并各自对极限进行了定义。其中，达朗贝尔的定义为：“若一个量能无限趋近于另一个量，且二者的差距小于任意给定的量，则前者称为后者的极限”，该定义已接近极限的正确定义；然而，这些人的定义均未能摆脱几何直观的束缚。出现这种情况是必然的，因为 19 世纪以前，算术与几何的大部分概念都建立在几何量概念的基础之上。

First of all, the definition of the derivative with the concept of limit is the correct definition of Czech mathematician Bolzano. He defined the derivative of the function $f(x)$ as the limit $f^{\prime }(x)$ of the difference quotient $\Delta y/\Delta x$ . He emphasized that $f^{\prime }(x)$ is not a quotient of two zeros [7]. Bolzano’s ideas are valuable, but he still hasn’t made clear about the nature of the limits.
首先，用极限概念定义导数的正确方式由捷克数学家波尔查诺（Bolzano）提出。他将函数 $f(x)$ 的导数定义为差商 $\Delta y/\Delta x$ 的极限 $f^{\prime }(x)$ ，并强调 $f^{\prime }(x)$ 并非两个零的商 [7]。波尔查诺的思想具有重要价值，但他仍未明确极限的本质。

By the 19th century, mathematics was plunged into huge contradictions. On the one hand, mathematics has made great achievements in describing and predicting physical phenomena. On the other hand, because a large number of mathematical structures have no logical basis, mathematics cannot be guaranteed to be correct. Advocated by German mathematicians, the mathematical community carried out a critical inspection campaign on mathematics, and rigorous definitions and rigorous proofs of some theories. French mathematician Cauchy was relatively complete on the basis of his predecessors’ work. He expounded the concept of limit and its theory. He pointed out in the “Analysis Tutorial”: “When the value that a variable takes sequentially successively approaches a fixed value, the difference between the value of the variable and the fixed value is as small as possible. This fixed value is called the limit value of all other values. In particular, when the value (absolute value) of a variable decreases infinitely to converge to the limit 0, it is said that the variable becomes infinitely small.”
到 19 世纪，数学陷入了巨大的矛盾之中。一方面，数学在描述和预测物理现象方面取得了丰硕成果；另一方面，由于大量数学结构缺乏逻辑基础，数学的正确性无法得到保证。在德国数学家的倡导下，数学界开展了对数学的批判性审视运动，致力于为部分理论建立严谨的定义与证明。法国数学家柯西（Cauchy）在前辈工作的基础上，较为系统地阐述了极限的概念及其理论。他在《分析教程》中指出：“当一个变量相继所取的值无限趋近于某一固定值，且变量的值与该固定值的差可以任意小时，这个固定值就称为所有其他值的极限值。特别地，当一个变量的绝对值无限减小并收敛于极限 0 时，就称这个变量为无穷小量。”

In order to eliminate the intuitive traces in the concept of limits, Weierstrass proposed a static definition of limits, which provided a strict theoretical basis for calculus. The so-called ${\lim }_{n\to \infty }{a}_n=A$ means: “If for any $\epsilon >0$ , there always exists a natural number $N$ , such that when $n>N$ , the inequality $|a_n-A|<\epsilon$ is constant.”
为消除极限概念中的直观痕迹，魏尔斯特拉斯（Weierstrass）提出了极限的静态定义，为微积分提供了严格的理论基础。所谓 ${\lim }_{n\to \infty }{a}_n=A$ ，是指：“对于任意 $\epsilon >0$ ，总存在自然数 $N$ ，使得当 $n>N$ 时，不等式 $|a_n-A|<\epsilon$ 恒成立。”

This definition, with the help of inequality, quantitatively and specifically characterizes the connection between the two “infinite processes” through the relationship between $\epsilon$ and $n$ . Therefore, this definition is strict and can be used as the basis for scientific demonstration. It is still used in mathematical analysis books. In this definition, only the number and its size relationship are involved. In addition, only the words such as given, existence, and optional have been rid of the word “approach” and no longer ask for help of intuitive for movement.
该定义借助不等式，通过 $\epsilon$ 与 $n$ 的关系，定量且具体地刻画了两个 “无限过程” 之间的联系，因此具有严格性，可作为科学论证的基础，至今仍在数学分析教材中使用。在这个定义中，仅涉及数及其大小关系，仅使用 “给定”“存在”“任意” 等词汇，摒弃了 “趋近” 这类表述，不再依赖运动的直观性。

After this, Weierstrass, Dedekind, and Cantor each independently and thoroughly researched the basis of analysis into the theory of real numbers, and each established a complete real number system in the 1870s. Weierstrass’s theory can be attributed to the principle of the existence of increasing bounded sequence numbers; Dedekind established the famous Dedekind segmentation; Cantor proposed the use of the limits of a rational “basic sequence” to define irrational numbers. They established a rigorous limit theory and real number theory, and completed the logical foundational work of analysis [8]. The inconsistency of mathematical analysis was summarized as the inconsistency of real number theory, thus making calculus that the unprecedented and magnificent mansion was built on a solid and reliable foundation. The important and difficult task of reconstructing the foundation of calculus was successfully completed by the efforts of many outstanding scholars. The perfection of the limit theory has made calculus solid basis.
此后，魏尔斯特拉斯、戴德金（Dedekind）与康托尔（Cantor）各自独立地深入研究分析学基础，并将其归结为实数理论，于 19 世纪 70 年代分别建立了完备的实数系。魏尔斯特拉斯的理论可归结为 “单调有界数列存在极限” 原理；戴德金创立了著名的 “戴德金分割” 理论；康托尔则提出用有理数 “基本序列” 的极限来定义无理数。他们建立了严谨的极限理论与实数理论，完成了分析学的逻辑奠基工作 [8]。数学分析的无矛盾性被归结为实数理论的无矛盾性，从而使微积分这座前所未有的宏伟大厦建立在坚实可靠的基础之上。重建微积分基础这一重要而艰巨的任务，在众多杰出学者的努力下得以顺利完成，极限理论的完善也为微积分奠定了稳固的基础。

3. Applications of Extreme Thoughts in Mathematical Analysis

3. 极限思想在数学分析中的应用

3.1 Application in Concepts

3.1 在概念中的应用

The method of thoughts of the limit runs through the course of mathematical analysis. It can be said that almost all concepts in mathematical analysis are inseparable from the limit. In almost all mathematical analysis books, the function theory and the method of thoughts of the limit are first introduced to give concepts of continuity, convergence and divergence of functions, derivatives, definite integrals, poles, double integrals, and curve integral and surface integral.
极限思想方法贯穿于数学分析课程的始终。可以说，数学分析中几乎所有概念的建立都离不开极限。在几乎所有的数学分析教材中，都会先介绍函数理论与极限思想方法，进而给出函数的连续性、敛散性、导数、定积分、极点、二重积分、曲线积分与曲面积分等概念。

(1) As for the continuous definition of a function $y=f(x)$ at the point $x_0$ : let $\Delta x=x-x_0$ be the increment or change amount of the independent variable $x$ at the point $x_0$ , set $y_0=f(x_0)$ , then the corresponding increment of the function $y$ at the point $x_0$ is $\Delta y=f(x)-f(x_0)=f(x_0+\Delta x)-f(x_0)=y-y_0$ . The function $y=f(x)$ is continuous at the point $x_0$ if and only if ${\lim }_{\Delta x\to 0}\Delta y=0$ , which means that when the increment $\Delta x$ of the independent variable $x$ approaches 0, the increment $\Delta y$ of the function value also approaches 0.
（1）函数 $y=f(x)$ 在点 $x_0$ 处的连续性定义：设 $\Delta x=x-x_0$ 为自变量 $x$ 在点 $x_0$ 处的增量（或改变量），令 $y_0=f(x_0)$ ，则函数 $y$ 在点 $x_0$ 处对应的增量为 $\Delta y=f(x)-f(x_0)=f(x_0+\Delta x)-f(x_0)=y-y_0$ 。函数 $y=f(x)$ 在点 $x_0$ 处连续，等价于 ${\lim }_{\Delta x\to 0}\Delta y=0$ ，即当自变量 $x$ 的增量 $\Delta x$ 趋近于 0 时，函数值的增量 $\Delta y$ 也趋近于 0。

(2) Definition of the derivative of function $y=f(x)$ at the point $x_0$ : if the function $y=f(x)$ is defined in a neighborhood of the point $x_0$ , and the limit ${\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ exists, then the function $f(x)$ is said to be differentiable at the point $x_0$ , and the limit is called the derivative of $f(x)$ at the point $x_0$ . Let $x=x_0+\Delta x$ , $\Delta y=f(x_0+\Delta x)-f(x_0)$ , then the derivative can be written as ${\lim }_{\Delta x\to 0}\frac{\Delta y}{\Delta x}={\lim }_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=f^{\prime }(x_0)$ . Therefore, the derivative is the limit of the ratio $\frac{\Delta y}{\Delta x}$ of the function increment $\Delta y$ to the independent variable increment $\Delta x$ .
（2）函数 $y=f(x)$ 在点 $x_0$ 处的导数定义：若函数 $y=f(x)$ 在点 $x_0$ 的某邻域内有定义，且极限 ${\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ 存在，则称函数 $f(x)$ 在点 $x_0$ 处可导，该极限称为 $f(x)$ 在点 $x_0$ 处的导数。令 $x=x_0+\Delta x$ ， $\Delta y=f(x_0+\Delta x)-f(x_0)$ ，则导数可表示为 ${\lim }_{\Delta x\to 0}\frac{\Delta y}{\Delta x}={\lim }_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=f^{\prime }(x_0)$ 。因此，导数是函数增量 $\Delta y$ 与自变量增量 $\Delta x$ 的比值 $\frac{\Delta y}{\Delta x}$ 的极限。

(3) Definition of definite integral of function $y=f(x)$ in the interval $[a,b]$ : let $f(x)$ be a function defined on $[a,b]$ , $J$ be a fixed real number. If for any given positive number $\epsilon$ , there always exists a positive number $\delta$ , such that for any partition $T$ of $[a,b]$ and any set of points { ξ i } \{\xi_{i}\} {ξi​} selected arbitrarily on each subinterval of $T$ , as long as $\Vert T\Vert <\delta$ (where $\Vert T\Vert$ is the mesh of the partition $T$ ), there is $|{\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i-J|<\epsilon$ , then $f(x)$ is said to be integrable on $[a,b]$ , and $J$ is called the definite integral of $f(x)$ on $[a,b]$ , denoted as $J={\int }_a^{b}f(x)dx$ . The definite integral is the limit of the Riemann sum ${\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i$ as the mesh of the partition approaches 0.
（3）函数 $y=f(x)$ 在区间 $[a,b]$ 上的定积分定义：设 $f(x)$ 是定义在 $[a,b]$ 上的函数， $J$ 是一个固定的实数。若对于任意给定的正数 $\epsilon$ ，总存在正数 $\delta$ ，使得对于 $[a,b]$ 的任意分割 $T$ 以及在 $T$ 的每个子区间上任意选取的点集 { ξ i } \{\xi_{i}\} {ξi​} ，只要 $\Vert T\Vert <\delta$ （其中 $\Vert T\Vert$ 为分割 $T$ 的细度），就有 $|{\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i-J|<\epsilon$ ，则称 $f(x)$ 在 $[a,b]$ 上可积， $J$ 称为 $f(x)$ 在 $[a,b]$ 上的定积分，记为 $J={\int }_a^{b}f(x)dx$ 。定积分是当分割细度趋近于 0 时，黎曼和 ${\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i$ 的极限。

(4) The convergence and divergence of a series ${\sum }_{n=1}^{\infty }{u}_n$ is defined by the limit of its partial sum sequence { S n } \{S_{n}\} {Sn​} (where $S_n={\sum }_{k=1}^n{u}_k$ ), and so on.
（4）级数 ${\sum }_{n=1}^{\infty }{u}_n$ 的敛散性由其部分和数列 { S n } \{S_{n}\} {Sn​} （其中 $S_n={\sum }_{k=1}^n{u}_k$ ）的极限来定义，诸如此类。

3.2 Application In Derivatives

3.2 在导数中的应用

The idea of derivative was originally introduced by French mathematician Fermat to study the extreme value problem, but directly related to the concept of derivative is the following two problems: Knowing the law of motion to find the speed and finding the tangent of the curve.
导数思想最初由法国数学家费马为研究极值问题而引入，但与导数概念直接相关的是以下两个问题：已知运动规律求速度，以及求曲线的切线。

(1) Instantaneous speed: Let a particle move along a straight line, and its motion law is $s=s(t)$ . Let $t_0$ be a certain moment, and $t$ be a moment adjacent to $t_0$ , then $\bar{v}=\frac{s(t)-s(t_0)}{t-t_0}$ is the average speed of the particle in the time interval $[t_0,t]$ . If the limit of the average speed $\bar{v}$ as $t\to t_0$ exists, then this limit ${\lim }_{t\to t_0}\frac{s(t)-s(t_0)}{t-t_0}$ is the instantaneous speed of the particle at the moment $t_0$ .
（1）瞬时速度：设一质点做直线运动，其运动规律为 $s=s(t)$ 。设 $t_0$ 为某一时刻， $t$ 为与 $t_0$ 相邻的时刻，则 $\bar{v}=\frac{s(t)-s(t_0)}{t-t_0}$ 为质点在时间区间 $[t_0,t]$ 内的平均速度。若当 $t\to t_0$ 时，平均速度 $\bar{v}$ 的极限存在，则该极限 ${\lim }_{t\to t_0}\frac{s(t)-s(t_0)}{t-t_0}$ 即为质点在时刻 $t_0$ 的瞬时速度。

(2) Slope of the tangent: The tangent line $PT$ of the point $P(x_0,y_0)$ on the curve $y=f(x)$ is the limit position of the secant line $PQ$ when the moving point $Q$ approaches the point $P$ infinitely along the curve.
（2）切线斜率：曲线 $y=f(x)$ 上点 $P(x_0,y_0)$ 处的切线 $PT$ ，是当动点 $Q$ 沿曲线无限趋近于点 $P$ 时，割线 $PQ$ 所趋近的极限位置。

Since the slope of the secant $PQ$ is $\bar{k}=\frac{f(x)-f(x_0)}{x-x_0}$ , so if the limit of $\bar{k}$ as $x\to x_0$ exists, then the limit $k={\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ is the slope of the tangent $PT$ .
由于割线 $PQ$ 的斜率为 $\bar{k}=\frac{f(x)-f(x_0)}{x-x_0}$ ，因此，若当 $x\to x_0$ 时， $\bar{k}$ 的极限存在，则该极限 $k={\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ 即为切线 $PT$ 的斜率。

Give the definition of the derivative: If the function $y=f(x)$ is defined in a neighborhood of the point $x_0$ , and the limit ${\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ exists, then we define that the function $f(x)$ is differentiable at the point $x_0$ , and the limit is the derivative of $f(x)$ at $x_0$ , denoted as $f^{\prime }(x_0)$ .
给出导数的定义：若函数 $y=f(x)$ 在点 $x_0$ 的某邻域内有定义，且极限 ${\lim }_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ 存在，则称函数 $f(x)$ 在点 $x_0$ 处可导，该极限即为 $f(x)$ 在 $x_0$ 处的导数，记为 $f^{\prime }(x_0)$ 。

Let $x=x_0+\Delta x$ , $\Delta y=f(x_0+\Delta x)-f(x_0)$ , then the above formula can be rewritten as ${\lim }_{\Delta x\to 0}\frac{\Delta y}{\Delta x}={\lim }_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=f^{\prime }(x_0)$ .
令 $x=x_0+\Delta x$ ， $\Delta y=f(x_0+\Delta x)-f(x_0)$ ，则上述导数定义式可改写为 ${\lim }_{\Delta x\to 0}\frac{\Delta y}{\Delta x}={\lim }_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=f^{\prime }(x_0)$ 。

3.3 Application in Integral

3.3 在积分中的应用

Integral is an important concept in mathematical analysis. The indefinite integral is the inverse operation of the derivative and the definite integral is the limit of some special sum [10]. The following gives the important application of the limit idea in definite integral.
积分是数学分析中的重要概念，其中不定积分是导数的逆运算，而定积分则是某种特殊和式的极限 [10]。以下阐述极限思想在定积分中的重要应用。

Background proposed by definite integral: A curved trapezoid is composed of a non-negative continuous curve $y=f(x)$ , straight lines $x=a$ , $x=b$ and the $x$ -axis. How to find the area of this curved trapezoid?
定积分的提出背景：由非负连续曲线 $y=f(x)$ 、直线 $x=a$ 、 $x=b$ 以及 $x$ 轴所围成的图形称为曲边梯形。如何求该曲边梯形的面积？

(1) Dividing the curved trapezoid into $n$ small curved trapezoids.
（1）将曲边梯形分割为 $n$ 个小曲边梯形。

(2) When $n$ is large and all $\Delta x_i(i=1,2,\cdots ,n)$ are very small, each small curved trapezoid can be approximated as a small rectangle. The area of the $k$ -th small curved trapezoid $\Delta S_k\approx f({\xi }_k)\Delta x_k(k=1,2,\cdots ,n)$ , where $x_{k-1}\le {\xi }_k\le x_k$ . At this time, the total area $S={\sum }_{k=1}^n\Delta S_k\approx {\sum }_{k=1}^{n}f({\xi }_k)\Delta x_k(k=1,2,\cdots ,n)$ .
（2）当 $n$ 很大且所有 $\Delta x_i(i=1,2,\cdots ,n)$ 都很小时，每个小曲边梯形可近似看作一个小矩形。第 $k$ 个小曲边梯形的面积 $\Delta S_k\approx f({\xi }_k)\Delta x_k(k=1,2,\cdots ,n)$ ，其中 $x_{k-1}\le {\xi }_k\le x_k$ 。此时，曲边梯形的总面积 $S={\sum }_{k=1}^n\Delta S_k\approx {\sum }_{k=1}^{n}f({\xi }_k)\Delta x_k(k=1,2,\cdots ,n)$ 。

(3) When $n$ increases infinitely, that is, when ∥ T ∥ = max ⁡ { Δ x 1 , Δ x 2 , ⋯   , Δ x n } \|T\|=\max \{\Delta x_{1}, \Delta x_{2}, \cdots, \Delta x_{n}\} ∥T∥=max{Δx1​,Δx2​,⋯,Δxn​} approaches 0 infinitely, the sum ${\sum }_{k=1}^{n}f({\xi }_k)\Delta x_k(k=1,2,\cdots ,n)$ approaches the area $S$ of the curved trapezoid infinitely. So $S={\lim }_{\Vert T\Vert \to 0}{\sum }_{k=1}^{n}f({\xi }_k)\Delta x_k$ .
（3）当 $n$ 无限增大，即分割细度 ∥ T ∥ = max ⁡ { Δ x 1 , Δ x 2 , ⋯   , Δ x n } \|T\|=\max \{\Delta x_{1}, \Delta x_{2}, \cdots, \Delta x_{n}\} ∥T∥=max{Δx1​,Δx2​,⋯,Δxn​} 无限趋近于 0 时，和式 ${\sum }_{k=1}^{n}f({\xi }_k)\Delta x_k(k=1,2,\cdots ,n)$ 无限趋近于曲边梯形的面积 $S$ 。因此， $S={\lim }_{\Vert T\Vert \to 0}{\sum }_{k=1}^{n}f({\xi }_k)\Delta x_k$ 。

Definite integral: Take $n-1$ points in the closed interval $[a,b]$ such that $a=x_0<x_1<\cdots <x_{n-1}<x_n=b$ . These points divide $[a,b]$ into $n$ small intervals ${\Delta }_i=[x_{i-1},x_i]$ ( $i=1,2,\cdots ,n$ ). These points or these closed subintervals constitute a partition of $[a,b]$ , denoted as T = { x 0 , x 1 , ⋯   , x n } T=\{x_{0}, x_{1}, \cdots, x_{n}\} T={x0​,x1​,⋯,xn​} or T = { Δ 1 , Δ 2 , ⋯   , Δ n } T=\{\Delta_{1}, \Delta_{2}, \cdots, \Delta_{n}\} T={Δ1​,Δ2​,⋯,Δn​} . The length of the $i$ -th small interval is $\Delta x_i=x_i-x_{i-1}$ , and the mesh of the partition $T$ is denoted as $\Vert T\Vert ={\max }_{1\le i\le n}\Delta x_i$ . Let $f(x)$ be a function defined on $[a,b]$ , $J$ be a fixed real number. If for any given positive number $\epsilon$ , there always exists a positive number $\delta$ , such that for any partition $T$ of $[a,b]$ and any set of points { ξ i } \{\xi_{i}\} {ξi​} selected arbitrarily on each subinterval of $T$ , as long as $\Vert T\Vert <\delta$ , there is $|{\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i-J|<\epsilon$ , then $f(x)$ is said to be integrable on $[a,b]$ , and $J$ is called the definite integral of $f(x)$ on $[a,b]$ , denoted as $J={\int }_a^{b}f(x)dx$ . It is the limit of the integral sum ${\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i$ when the segmentation fineness approaches to zero.
定积分的定义：在闭区间 $[a,b]$ 内取 $n-1$ 个点，使得 $a=x_0<x_1<\cdots <x_{n-1}<x_n=b$ 。这些点将 $[a,b]$ 分割为 $n$ 个小区间 ${\Delta }_i=[x_{i-1},x_i]$ （ $i=1,2,\cdots ,n$ ）。这些点或这些闭子区间构成了对 $[a,b]$ 的一个分割，记为 T = { x 0 , x 1 , ⋯   , x n } T=\{x_{0}, x_{1}, \cdots, x_{n}\} T={x0​,x1​,⋯,xn​} 或 T = { Δ 1 , Δ 2 , ⋯   , Δ n } T=\{\Delta_{1}, \Delta_{2}, \cdots, \Delta_{n}\} T={Δ1​,Δ2​,⋯,Δn​} 。第 $i$ 个小区间的长度为 $\Delta x_i=x_i-x_{i-1}$ ，分割 $T$ 的细度记为 $\Vert T\Vert ={\max }_{1\le i\le n}\Delta x_i$ 。设 $f(x)$ 是定义在 $[a,b]$ 上的函数， $J$ 是一个固定的实数。若对于任意给定的正数 $\epsilon$ ，总存在正数 $\delta$ ，使得对于 $[a,b]$ 的任意分割 $T$ 以及在 $T$ 的每个子区间上任意选取的点集 { ξ i } \{\xi_{i}\} {ξi​} ，只要 $\Vert T\Vert <\delta$ ，就有 $|{\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i-J|<\epsilon$ ，则称 $f(x)$ 在 $[a,b]$ 上可积， $J$ 称为 $f(x)$ 在 $[a,b]$ 上的定积分，记为 $J={\int }_a^{b}f(x)dx$ 。它是当分割细度趋近于 0 时，积分和 ${\sum }_{i=1}^{n}f({\xi }_i)\Delta x_i$ 的极限。

4. Conclusion

4. 结论

Extreme thought is the basis of calculus. The limit method provides a powerful tool for humans to understand infinity. It prominently shows the characteristics of calculus different from elementary mathematics from the methodology. It is an important idea of modern mathematics. Extreme thought contains rich dialectic thoughts and is an excellent application of the unified laws of opposition of materialist dialectics in the field of mathematics. It will have a certain theoretical significance for some problems in the history of mathematics and the philosophy of mathematics to clear the development context of extreme thoughts. It reveals the core content of extreme thoughts and its internal connection with philosophical thoughts. It has an excellent promotion role in cultivating people’s thinking methods, thinking qualities, and improving their ability to analyze and solve problems.
极限思想是微积分的基础。极限方法为人类认识 “无穷” 提供了有力工具，从方法论层面凸显了微积分区别于初等数学的特点，是现代数学的重要思想。极限思想蕴含丰富的辩证思想，是唯物辩证法对立统一规律在数学领域的出色应用。理清极限思想的发展脉络，对于解决数学史与数学哲学中的部分问题具有一定理论意义；它揭示了极限思想的核心内容及其与哲学思想的内在联系，对培养人们的思维方式、思维品质，以及提升分析和解决问题的能力具有良好的促进作用。

References

参考文献

[1] Wang Zhipan. Application of extreme thoughts in mathematics [J]. Curriculum Education Research. 2015 (35): 125-127.
[1] 王志攀。极限思想在数学中的应用 [J]. 课程教育研究，2015 (35): 125-127.

[2] Wang Zhenfu. Exploring the calculation method of limit [J]. Mathematics Learning and Research. 2016 (23): 56-58.
[2] 王正福。极限的计算方法探究 [J]. 数学学习与研究，2016 (23): 56-58.

[3] Jin Haitao. Discussion on teaching practice of the limits of higher mathematics [J]. Mathematics Learning and Research. 2017 (05): 378-379.
[3] 金海淘。高等数学极限的教学实践探讨 [J]. 数学学习与研究，2017 (05): 378-379.

[4] Wang Juan. Infiltration of philosophical thoughts in calculus teaching [J]. Journal of Higher Correspondence Education (Natural Science Edition). 2007 (06): 202-205.
[4] 王娟。微积分教学中哲学思想的渗透 [J]. 高等函授学报（自然科学版）, 2007 (06): 202-205.

[5] Jiang Yukun. Application of extreme ideas in solid geometry [J]. Reference for Mathematics Teaching in Middle School. 2018 (36): 94-96.
[5] 姜玉坤。极限思想在立体几何中的应用 [J]. 中学数学教学参考，2018 (36): 94-96.

[6] Yang Zhuowen. Research on the application of extreme thought in problem solving [J]. Economic and Trade Practices. 2018 (02): 153-155.
[6] 杨卓文。极限思想在解题中的应用研究 [J]. 经贸实践，2018 (02): 153-155.

[7] Wang Hongjun, Jia Yuexian. Extreme thoughts in calculus and their applications [J]. Science and Technology Innovation Herald. 2017 (33): 189-192.
[7] 王洪军，贾月仙。微积分中的极限思想及其应用 [J]. 科技创新导报，2017 (33): 189-192.

[8] Xu Xiuheng. Seeing the magic in the ordinary–Study on the construction of extreme thoughts [J]. Mathematics Learning and Research (Teaching and Researching Edition). 2007 (03): 43-46.
[8] 徐秀恒。于平凡中见神奇 —— 极限思想的构建研究 [J]. 数学学习与研究（教研版）, 2007 (03): 43-46.

[9] Xie Huijie. The emergence, development and improvement of extreme ideas [J]. Mathematics Learning and Research (Teaching and Researching Edition). 2008 (09): 287-290.
[9] 谢慧杰。极限思想的产生、发展与完善 [J]. 数学学习与研究（教研版）, 2008 (09): 287-290.

[10] Gong Qunqiang. On the importance of “extreme thoughts” in teaching [J]. Mathematics Learning and Research. 2010 (13): 323-325.
[10] 龚群强。论 “极限思想” 在教学中的重要性 [J]. 数学学习与研究，2010 (13): 323-325.

参考文献

[1] Wang Zhipan. Application of extreme thoughts in mathematics [J]. Curriculum Education Research. 2015 (35): 125-127.
[1] 王志攀. 极限思想在数学中的应用[J]. 课程教育研究, 2015 (35): 125-127.

[2] Wang Zhenfu. Exploring the calculation method of limit[J]. Mathematics Learning and Research. 2016 (23): 56-58.
[2] 王正福. 极限的计算方法探究[J]. 数学学习与研究, 2016 (23): 56-58.

[3] Jin Haitao. Discussion on teaching practice of the limits of higher mathematics[J]. Mathematics Learning and Research. 2017 (05): 378-379.
[3] 金海淘. 高等数学极限的教学实践探讨[J]. 数学学习与研究, 2017 (05): 378-379.

[4] Wang Juan. Infiltration of philosophical thoughts in calculus teaching[J]. Journal of Higher Correspondence Education (Natural Science Edition). 2007 (06): 202-205.
[4] 王娟. 微积分教学中哲学思想的渗透[J]. 高等函授学报（自然科学版）, 2007 (06): 202-205.

[5] Jiang Yukun. Application of extreme ideas in solid geometry[J]. Reference for Mathematics Teaching in Middle School. 2018 (36): 94-96.
[5] 姜玉坤. 极限思想在立体几何中的应用[J]. 中学数学教学参考, 2018 (36): 94-96.

[6] Yang Zhuowen. Research on the application of extreme thought in problem solving[J]. Economic and Trade Practices. 2018 (02): 153-155.
[6] 杨卓文. 极限思想在解题中的应用研究[J]. 经贸实践, 2018 (02): 153-155.

[7] Wang Hongjun, Jia Yuexian. Extreme thoughts in calculus and their applications[J]. Science and Technology Innovation Herald. 2017 (33): 189-192.
[7] 王洪军, 贾月仙. 微积分中的极限思想及其应用[J]. 科技创新导报, 2017 (33): 189-192.

[8] Xu Xiuheng. Seeing the magic in the ordinary–Study on the construction of extreme thoughts[J]. Mathematics Learning and Research (Teaching and Researching Edition). 2007 (03): 43-46.
[8] 徐秀恒. 于平凡中见神奇——极限思想的构建研究[J]. 数学学习与研究（教研版）, 2007 (03): 43-46.

[9] Xie Huijie. The emergence, development and improvement of extreme ideas[J]. Mathematics Learning and Research (Teaching and Researching Edition). 2008 (09): 287-290.
[9] 谢慧杰. 极限思想的产生、发展与完善[J]. 数学学习与研究（教研版）, 2008 (09): 287-290.

[10] Gong Qunqiang. On the importance of “extreme thoughts” in teaching[J]. Mathematics Learning and Research. 2010 (13): 323-325.
[10] 龚群强. 论“极限思想”在教学中的重要性[J]. 数学学习与研究, 2010 (13): 323-325.

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吴文忠：极限（Limit）发展的前世今生[教学研究]

Feb 26, 2015 From：Original Author：吴文忠

极限是微积分学的核心概念之一，极限理论的完善得益于 19 世纪法国数学家柯西（Cauchy）和德国数学家魏尔斯特拉斯（Weierstrass）等的卓越工作。

今天，我们能相对“容易地”理解微积分，也得益于我们站在 Cauchy 等“巨人”肩膀上学习微积分，真心感谢他们对数学的贡献。向前追溯到两千五百年，极限思想的萌芽阶段，我们不得不提到古希腊的阿基米德（Archimedes），中国的惠施（Hui Shi）、刘徽（Liu Hui）、祖冲之（Tsu Chung-Chi）等数学家。

公元前 5 世纪，距离我们今天站立的时代约 2500 年，古希腊数学家安蒂丰（Antiphon）提出了“穷竭法”（Method of Exhaustion）。之后，古希腊数学家欧多克斯（Eudoxus）进一步完善“穷竭法”，使其成为一种合格的几何方法；再过 100 多年，阿基米德（Archimedes）做了进一步发展，其著作《论球和圆柱》运用“穷竭法”建立命题：只要边数足够多，圆外切正多边形的面积与内接正多边形的面积之差可以任意小。

以现代数学来看，阿基米德（Archimedes）建立的命题饱含“极限思想”，用无限逼近的方式从有限中认识无限、从近似中认识精确，已经十分贴近现代对于极限的认识。遗憾的是，古希腊人对数学公理化、机械化的追求，使其对“无限”似乎抱有“恐惧”，极力避免明显地人为“取极限”，转而用“归谬法”（Reduction to Absurdity）实现相关证明，搁置“极限”，导致极限理论（Theory of Limit）发展止步不前。

几乎同一时代，当阿基米德（Archimedes）在浴缸中探索“浮力定律”而非享受热水澡时，距离爱琴海岸 10000 多公里外的东方古国——中国，正处于春秋战国时期的社会大变革中，各类思想激烈碰撞、群星闪烁、百家争鸣。其中，名家学派创始人惠施是发展“极限思想”的一颗闪耀明星，他提出“一尺之锤，日取其半，万世不竭”，暗合“极限思想”的核心要义。

“一尺之锤，日取其半，万世不竭”的含义是：“一尺的木杖，今天取它的一半，明天取剩下部分的一半，后天再取第二天剩余部分的一半，始终会有一半剩余，因此永远取之不尽”。以现代数学视角来看，这是数列极限（Limit of a Sequence）概念的形象表达。约 500 年后的魏晋南北朝时期，数学家刘徽（Liu Hui）和祖冲之（Tsu Chung-Chi）将“极限的应用”推向了新高度。

刘徽为计算圆周率 $\pi$ ，以极限思想为指导提出“割圆术”（Cyclotomic Method），即通过圆内接正多边形的面积无限逼近圆的面积，进而求解圆周率 $\pi$ 。祖冲之（Tsu Chung-Chi）在刘徽“割圆术”的基础上，进一步求得约率 $\pi \approx \frac{22}{7}$ 及密率 $\pi \approx \frac{355}{113}$ 。在圆周率 $\pi$ 的计算领域，祖冲之应用“极限”（Limit）的成果领先西方数学 1000 余年。

在数学分析（Mathematical Analysis）出现之前，割圆术在圆周率计算史上长期被使用。若用三角函数（Trigonometric Function）表述，其关系为： $\pi \approx \cos \left(90^{\circ }-\frac{180^{\circ }}{n}\right)\times n$ 。当 $n$ 越大时， $\cos \left(90^{\circ }-\frac{180^{\circ }}{n}\right)\times n$ 越接近 $\pi$ ，写成极限形式即为：
$$\pi =\underset{n\to \infty }{\lim }\cos \left(90^{\circ }-\frac{180^{\circ }}{n}\right)\times n$$

深入探究古希腊的“穷竭法”（Method of Exhaustion）与古中国的“割圆术”（Cyclotomic Method）会发现，追溯至 2500 余年前，中西方数学以各自的发展轨迹萌芽极限思想，却又惊人地不谋而合，用独特的方式展现了“极限思想”的魅力。这不禁引发思考：中西方数学均发端于“极限思想”，为何未能进一步发展出“极限理论”便戛然而止？

或许原因在于：中国古代数学未形成严密的逻辑演绎体系，极限理论缺乏生长的沃土，“割圆术”最终止步于“求圆周率”这一具体应用；而在西方，爱琴海岸的文明虽延续，阿基米德（Archimedes）的智慧却随其生命终结而被中断——罗马士兵砍下了他的头颅。公元 3 世纪末，亚历山大图书馆毁于战火；公元 5 世纪末，欧洲文明陷入漫长而黑暗的中世纪，极限理论的发展也随之停滞。

朝代更迭、四季轮转，月球上的“祖冲之环形山”孤独地守护着地球，等待“极限理论”发展的曙光……而这束曙光，终将在时光的流转中到来。

当“祖冲之环形山”围绕地球孤独绕行约 12000 圈后，公元 14 世纪的欧洲掀起文艺复兴（Renaissance）思潮。但丁（Dante）、薄伽丘（Boccaccio）、达·芬奇（Da Vinci）、莎士比亚（Shakespeare）等时代巨匠带领人们逐渐冲破“神学”的思想禁锢；与此同时，麦哲伦（Magellan）、哥伦布（Columbus）等航海家怀着对财富的探索欲，开启了地理大发现（Age of Exploration）时代。

当莎士比亚笔下的朱丽叶拔剑倒在罗密欧的血泊中时，哥伦布已踏上印第安人世代居住的土地。此时，古希腊、古印度、古中国等文明积累的初等数学（Elementary Mathematics）逐渐无法满足时代需求，数学开始向更高阶段迈进。

欧洲文艺复兴与地理大发现时期，中国正处于明清时期。受理学盛行、八股取士等因素影响，中国数学发展缓慢，最终与微积分（Calculus）及极限理论的发展失之交臂。约 300 年后的公元 17 世纪，一方面，大量科学问题亟待解决；另一方面，费马（Fermat）、笛卡尔（Descartes）、开普勒（Kepler）、卡瓦列利（Cavalieri）、巴罗（Barrow）等数学家积累了丰富成果，英国数学家牛顿（Newton）与德国数学家莱布尼茨（Leibniz）在此基础上分别创立了微积分。

微积分（Calculus）的诞生不仅推动数学空前繁荣，更超越数学领域，在物理、化学、生物学、航海、天文学、机械制造、工程学、军事等众多领域广泛应用并取得巨大成就。然而，牛顿（Newton）与莱布尼茨（Leibniz）创立的微积分缺乏严谨的逻辑基础——人类进入数学发展的更高阶段时，伴随而来的是“已死量的幽灵”。这个“幽灵”质疑微积分的研究对象、原则及论断是否比宗教神秘主义、信仰教义具有更清晰的表达或更严谨的推理，最终引发“第二次数学危机”。

此后不到 100 年，公元 18 世纪 60 年代，第一次工业革命（The First Industrial Revolution）爆发，以蒸汽机的广泛应用为标志，人类迎来技术发展史上的重大变革。在此背景下，无论是数学自身的发展需求，还是人类技术进步的现实需要，都迫切要求微积分实现严谨化，而极限理论正是实现这一目标的关键。

公元 18 世纪，一批数学家为极限与微积分的深度融合做出卓有成效的贡献，包括达朗贝尔（D’Alembert）、欧拉（Euler）、拉格朗日（Lagrange）等。其中，达朗贝尔（D’Alembert）定性给出极限的定义：“一个变量趋于一个固定量，其趋于的程度小于任何给定的量，且变量永远无法达到固定量”。此外，他还运用极限思想提出判定级数（Series）敛散性的“达朗贝尔判别法”（D’Alembert’s Test）：

对于正项级数 ${\sum }_{n=1}^{\infty }{a}_n$ ，若 ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|=\rho$ ，则有以下结论：

1. 当 $\rho <1$ 时，级数收敛；
2. 当 $\rho >1$ 时，级数发散；
3. 当 $\rho =1$ 时，级数可能收敛也可能发散。

在现代数学中，达朗贝尔判别法（D’Alembert’s Test）仍是判定级数敛散性的重要方法。同一时期，欧拉（Euler）极大地拓展了微积分涉及的函数类型及求解方法——事实上，牛顿（Newton）与莱布尼茨（Leibniz）的研究仅覆盖少量函数及求法。在极限理论发展方面，欧拉（Euler）提出关于无穷小量的“不同阶零”理论。

此外，拉格朗日（Lagrange）专注于数学分析研究，是仅次于欧拉的数学分析开拓者。尽管拉格朗日回避极限概念，但他承认微分法可建立在极限理论的基础上。这一阶段，真正意义上的极限定义开始形成，不过其表述仍过于直观，与数学追求的严谨性原则存在冲突。

公元 19 世纪是数学严格性高度发展的时代，随着数学分析的深入推进，极限理论最终为微积分注入严谨性。在此过程中，波尔查诺（Bolzano）、柯西（Cauchy）、阿贝尔（Abel）、魏尔斯特拉斯（Weierstrass）等数学家为极限理论的最终创立做出杰出贡献。

首先，捷克数学家波尔查诺（Bolzano）摒弃无穷小量概念，运用极限定义导数与连续性，并得到判定级数收敛的一般准则——柯西准则（Cauchy’s Convergence Test）。遗憾的是，波尔查诺的工作长期被埋没，未对当时的数学发展产生影响。

随后，法国数学家柯西（Cauchy）发表《分析教程》，独立得出波尔查诺此前证明的基本结论，并以极限为基础定义无穷小量及微积分学的基本概念，建立级数收敛性的一般理论，成为对分析严格化影响最深远的数学家。不过，以现代标准衡量，柯西的分析理论多基于几何直观，严谨性仍有待提升。即便如此，柯西与当时被他忽视的挪威青年数学家阿贝尔（Abel），共同极大地推动了数学分析的严格化进程。

此后，德国数学家魏尔斯特拉斯（Weierstrass）构造出“魏尔斯特拉斯函数”（Weierstrass Function）——一个由无穷级数定义的函数。从直观上看，它是一条连续的锯齿状折线，且锯齿的尺寸无限缩小。这一函数的提出，否定了当时数学家“除少数特殊点外，连续函数处处可导”的普遍认知。随后，狄利克雷函数（Dirichlet Function）、黎曼函数（Riemann Function）、赫维赛德函数（Heaviside Function）等“病态函数”的出现，进一步证明直观与几何思维在数学研究中的局限性。

最终，魏尔斯特拉斯在前期数学家工作的基础上，定量给出极限的严格定义——即现代数学通用的“ $\epsilon -\delta$ 定义”：

对任意 $\epsilon >0$ ，存在 $\delta >0$ ，使得当 $0<|x-x_0|<\delta$ 时，恒有 $|f(x)-A|<\epsilon$ ，记为：
$$\underset{x\to x_0}{\lim }f(x)=A$$

魏尔斯特拉斯以极限理论为基础，严格建立微积分体系，系统创立实分析与复分析，基本实现“分析的算术化”，成功克服数学发展中的危机与矛盾，因此被尊为“现代分析之父”。而这一严格化的微积分体系，也为 20 世纪数学的发展奠定了坚实基础。

参考文献：
[1] 王青建，武修文，王玥. 古希腊数学演绎证明思想的发展脉络[J]. 辽宁师范大学学报（自然科学版），2003
[2] Peter Beckmann. 《A History of $\pi$ 》[M]. Boulder, Colo: Golem Press，1977
[3] 庄周. 《庄子·天下篇》
[4] 李玉玲. 极限概念的建立及其重要作用[J]. 宝鸡文理学院学报（社会科学版），2012
[5] 吴文忠. AP 微积分辅导手册[M]. 北京：化学工业出版社，2018
[6] 陈宇. 极限论的发展[J]. 邯郸大学学报，2000
[7] 李心灿. 微积分的创立者及其先驱（修订版）[M]. 北京：高等教育出版社，2002
[8] 项晶菁. 数学文化选粹[M]. 西安：西北大学出版社，2015
[9] 人民教育出版社历史室. 《世界近代现代史》[M]. 北京：人民教育出版社，2000
[10] 凌国英. 关于达朗贝尔判别法[J]. 湖州师范学院学报
[11] J.A 塞雷（Serret）. 《拉格朗日文集》
[12] 李文林. 数学珍宝[M]. 北京：科学出版社，1998
[13] 李后强，汪富泉. 分形理论及其发展历程[D]，1992
[14] 苏维钢，钟怀杰. 实分析中某些病态函数的性质和作用

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via：

• G6881-MCANS-153-数学 - Tang_2020_J._Phys.:_Conf._Ser._1650_032204.pdf
  https://iopscience.iop.org/article/10.1088/1742-6596/1650/3/032204/pdf

• 吴文忠：极限（Limit）发展的前世今生[教学研究]-教材/真题-AP微积分网
  https://calculus.apexams.net/20/2021.html