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极限思想及其在数学中的应用

摘 要：高等数学中极限教学作为重要内容，是高等数学计算分析的基础，也是高等数学问题分析的难题，极限的基本思考都是围绕高等数学计算分析开展的，高等数学中微积分、级数等基础概念和思想都是基于极限思想提出的，以极限作为工具去解决和处理数学问题是一种极其重要的方法。许多学生在学习数列极限时感觉很困难，原因在于数列极限概念很抽象，而且计算也有一定的难度。本文首先阐述极限的定义；接着从数列极限和函数极限两方面分析极限的求解方法；最后指出极限的应用状况，通过这些应用使我们对极限有一个更系统立体的了解。

关键词：极限；求解方法；应用状况

Limit thought and its application in mathematics

Abstract：Limits in higher mathematics teaching as an important content, is the foundation of higher

mathematics calculation and analysis, is also a difficult problem in higher mathematics problem analysis, limit the basic thinking about higher mathematics calculation and analysis, calculus of higher mathematics, series, and other basic concepts and ideas are put forward based on the limit state, in order to limit as a tool to solve and deal with the mathematics problem is a very important method. Many students find it difficult to learn the limit of the sequence because the concept of the limit is abstract and computationally difficult. Firstly, the definition of limit is described. Then the solution method of limit is analyzed from the limit of sequence and the limit of function. Finally, the application of the limit is pointed out. Through these applications, we have a more systematic understanding of the limit.

Key words: limit; Solution method; Application status

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目 录

一、引言.............................................................................................................................................. 1

（一） 选题背景 ................................................................................................................... 1 （二）研究目的和意义 ........................................................................................................ 1 二、极限的概念 ............................................................................................................................... 1

（一）数列极限的定义 ........................................................................................................ 1 （二）函数极限的定义 ........................................................................................................ 2

1 一元函数极限的定义 ............................................................................................... 2 2 多元函数极限的定义 ............................................................................................... 3

三、极限的求法 ............................................................................................................................... 4

（一） 数列极限的求法 ...................................................................................................... 4

1 极限定义求法 ............................................................................................................ 4 2 极限运算法则法 ........................................................................................................ 7 3 夹逼准则求法 ............................................................................................................ 7 4 单调有界定理求法 ................................................................................................... 8 5 定积分定义法 ............................................................................................................ 8 6 级数法 .......................................................................................................................... 9 （二）函数极限的求法 ........................................................................................................ 9

1 一元函数极限的求解方法 ...................................................................................... 9 2 多元函数极限的求解方法 .................................................................................... 16

四、极限的应用 ............................................................................................................................. 19

（一）在计算面积中的应用 ............................................................................................. 19

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（二）在求方程数值解中的应用 .................................................................................... 20 五、结论 .......................................................................................................................................... 21 致 谢 ............................................................................................................................................... 23

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