极限理论在数学分析中地位作用.doc

极限理论在数学分析中地位与作用 摘要 极限理论是数学分析的基本理论，极限概念是极限理论的核心。极限具有重大意义The status and role of the limit theory in mathematical analysis ABSTRACT Limit theory is a mathematical analysis of the basic theory, the concept of limit is the core of the theory of limit., This article from the continuous derivative, the definite integral, and the convergence of the series such as solving the limit, the limit problem involved in all aspects of in-depth exploration.. First of all start from the definition, find the limit of a function with the series limit contact, and thus the use of the limits of nature, criteria, Cauchy limit theory, forcing convergence method for solving different types of function, sequence limit. Solving the limit on the basis of the limit defined that infinitesimal substitution method for solving the limit. While examples of the types of special limit to get the solution calculated, summed up a number of important laws and relevant conclusions. These conclusions are laid limit the status and role of theory in mathematical analysis, to better discrimination method and more complete theoretical system limit for the subsequent study and research, mathematical analysis is of great significance. Key words: The limit, ordered series of numbers, Function, Definite integral, Determine the conditions 目录 1 引言 4 2 极限的思想渊源与发展史 5 2.1 极限的思想及历史 5 2.1.1 最早的无限分割思想 6 2.1.2. 西方的穷竭法与中国的割圆术 6 2.1.3. 斯杰文对穷竭法的简化和瓦里斯的算术化 9 3 极限的相关理论 10 3.1 极限概念的逐步形成 10 3.2 极限概念的完善 10 3.2.1函数极限 12 3.2.2数列极限 15 3.3 极限理论的确立 16 3.3.1波尔查诺的工作 16 3.3.2 柯西的极限理论 17 3.3.3 维尔斯特拉斯的静态理论 17 4 数学分析中极限的作用 17 4.1 函数的连续 19 4.2 数列的收敛性 21 4.2.1 唯一性 21 4.2.2 有界性 22 4.3 导数是特殊的极限 22 5 极限的计算 24 5.1 利用导数的定义 24 5.2 利用初等函数的连续性 24 5.3 数列极限 25 5.3.1 利用函数极限求数列极限 25 5.3.2 利用定积分求数列极限 25 5.4 函数极限 26 5.4.1 利用迫敛性求函数极限 26 5.4.2利用罗比达法则求函数极限 26 5.4.3 利用泰勒级数展开式求函数极限 27 5.4.4 利用中值定理求函数极限 27 5.4.5 利用定积分的定义求函数极限 28 5.4.6 利用等价无穷小替换求函数极限 28 5.4.7 利用收敛级数的必要条件求函数极限 28 5.5利用级数收敛的必要条件求极限 28 5.6.将数列的极限化为定积分 29 6 结论 30 7 参考文献 30 致谢 31 1 引言 数学分析课程的是人类二十世纪的