函数极限的计算方法研究

北方民族大学

学士学位论文

论文题目:函数极限的计算方法研究

院(部)名 称: 信 息 与 计 算 科 学 学 生 姓 名: 袁 雪 华 专 业:数学与应用数学 学 号:20093212 指导教师姓名: 万 仁 霞 论文提交时间: 2012年05月01日 论文答辩时间: 学位授予时间:

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函数极限的计算方法研究

摘 要

函数极限的计算方法是近代微积分学的基础知识，是学好高等数学的入门课题。本文主要探讨了一元函数的极限的基本性质、以及在此基础上完成的函数极限的利用夹逼性，等价代换和已知极限求极限等方法。进行函数极限的计算方法有很多种，例如利用洛比达法则求函数极限，泰勒展开求函数极限，利用函数连续性求极限以及利用积分中值定理求极限等都是常用且简洁的方法。每种方法都有其使用的条件以及其优缺点，对于具体问题应该做到具体分析，进而选择最简单最有效的方法进行求解。

关键字：函数极限， 等价代换， 积分中值定理，Taylor公式

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函数极限的计算方法研究

Abstract

Calculation Method of function limit is the foundation of modern calculus knowledge, is getting started to learn Higher Mathematics subject. This article mainly discusses the basic nature of the limit of a function, as well as done on the basis of this function to limit the use of clamping force and equivalent substitution and known methods such as extreme limit.There are many kinds of methods of calculation of the limit of function, such as the limit of function using Los law of Breda, a Taylor expansion function limit, use the continuity of function limit and limit the use of integral mean value theorem are common and simple way.Each method has its conditions of use, as well as its advantages and disadvantages, for specific issues should be specific points.

key word:The limit function, Equivalent substitution, Integral mean value theorem , Taylor formula

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函数极限的计算方法研究

目 录

第一章 引 言.................................................................................................................................... 1 第二章 函数极限...................................................................................................................................... 2

2.1基本概念...................................................................................................................................... 2

2.1.1 函数极限的定义 .............................................................................................................. 2 2.1.2 函数极限的几何意义 ...................................................................................................... 2 2.2. 函数极限的分类 ........................................................................................................................ 3

2.2.1 单侧极限 .......................................................................................................................... 3 2.2.2 函数在无穷远处的极限 .................................................................................................. 3 2.3函数极限存在的准则 .................................................................................................................. 4

2.3.1归结原则 ........................................................................................................................... 4 2.3.2柯西准则 ........................................................................................................................... 5 2.4函数极限的性质 .......................................................................................................................... 5

2.4.1极限的唯一性 ................................................................................................................... 5 2.4.2局部保序性 ....................................................................................................................... 6 2.4.3夹逼性 ............................................................................................................................... 7 2.4.4函数极限的四则运算 ....................................................................................................... 8 2.5本章小结...................................................................................................................................... 8 第三章 利用等价代换和初等变形求极限 ............................................................................................ 9

3.1.等价代换...................................................................................................................................... 9 3.2利用初等变形求函数极限 ........................................................................................................ 11 3.3本章小结.................................................................................................................................... 11 第四章 利用已知极限求函数极限 ........................................................................................................ 12

4.1两个著名极限 ............................................................................................................................ 12 4.2利用已知极限求极限推广 ........................................................................................................ 13 4.3本章小结.................................................................................................................................... 14 第五章 几种特殊求极限的方法 ............................................................................................................ 15

5.1 L’Hospital法则 .......................................................................................................................... 15 5.2用Taylor公式求极限 ............................................................................................................... 16

5.2.1 Taylor公式 ..................................................................................................................... 16 5.2.2用Taylor公式求极限 .................................................................................................... 17 5.3 利用积分定义求极限 ............................................................................................................... 18 5.4利用连续性求函数极限 ............................................................................................................ 19 5.5利用积分中值定理求极限 ........................................................................................................ 20 5.6本章小结.................................................................................................................................... 21 结 束 语 ................................................................................................................................................. 22 致 谢 ..................................................................................................................................................... 23 参考文献 ................................................................................................................................................. 24 附录一 英文原文.................................................................................................................................... 25

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