Heine定理及其应用

目录

1引言 (1)

2文献综述 (1)

2.1 国外研究现状 (1)

2.2 国内研究现状 (1)

2.3 国内外研究现状评价 (2)

2.4 提出的问题 (2)

3 Heine定理及其不同结论 (2)

3.1 Heine定理的证明 (2)

3.2 Heine定理的推广 (4)

4 Heine定理的应用 (6)

4.1 判断、证明函数极限的存在性 (6)

4.2 利用Heine定理求极限 (8)

4.2.1 求函数极限 (8)

4.2.2 求数列极限 (8)

4.3 证明函数极限的性质 (9)

4.4 判断函数在某点的可导性 (11)

4.5 判断级数敛散性 (12)

4.6 对函数()x f的局部利用海涅定理，求函数()x f的极限 (13)

4.7 根据函数的特性，应用海涅定理分析解决其他问题 (14)

5 总结 (16)

5.1 主要发现 (16)

5.2 启示 (17)

5.3 局限性 (17)

5.4 努力方向 (17)

参考文献 (18)

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Heine定理及其应用

摘要

Heine定理又称为归结原理，是工科数学分析和高等数学中判断数列极限和函数极限存在的一种有效的重要方法。它是分析学中的重点和难点，在极限理论中发挥了重要的作用。国内外有关Heine定理的若干问题的探讨和应用研究非常多，涉及范围很广，说明了其重要性和应用的广泛性。国外对Heine定理的研究主要是解决函数和数列极限存在性问题中的应用，在教学上探讨理论应用涉及甚少，而国内在其理论方面的研究甚为广泛，但Heine定理的定义及应用仍有值得研究的问题。比如：Heine定理通常用于极限的存在性问题，而其用途不仅仅限于此，但由于Heine定理的充分较强，使得Heine 定理在应用中存在着一定的局限性，是否能够将Heine定理的充分性条件进一步弱化，使得在用Heine定理处理极限理论问题时更加实用方便，以及在判断级数敛散性、证明函数性质定理、函数求导问题中的应用，这就是文章探讨的问题所在，这样的研究在国内外相对较少。基于已有的Heine定理若干定义，该文在前人研究的基础上，结合数学分析中的相关定义及性质，分析了Heine定理的内涵及其推论和相应的证明，分别给出了不同形式的Heine定理，并对Heine定理做出了相应的推广，并利用Heine定理和数列极限的性质，探讨函数极限的相关性质，以及判断函数极限是否存在，给出了详细的实例，通过实例分析Heine定理在优化极限判断及运算和其他方面的应用，体现了Heine 定理在判断数列和函数极限存在及研究函数性质问题、级数敛散性判断问题中的优越性和实用性。

关键词：Heine定理；数列；函数；极限；推广；应用

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Heine theorem and its application

Abstract：Heine theorem is also known as the resolution principle, is the analysis of Engineering Mathematics and higher mathematics in the judgement of sequence limits and function limit there is a kind of effective method. It is the analysis of key point and the difficulty, in the limit theory played an important role in. Domestic and foreign related Heine theorem on the study and application of very much, involve range is very wide, illustrates its importance and wide application. On Heine theorem research mainly solves the problem of the existence of the limit of a sequence of function and application in the teaching of theory and application, very involved, and in the domestic theoretical research is very widely, but the Heine theorem and application still has a problem worthy of study. For example: Heine theorem is usually used to limit existence problem, but its use is not limited to this, but since Heine theorem sufficient strong, making Heine theorem in the application of certain limitations exist, whether can be Heine theorem sufficient conditions for further weakening, the Heine theorem of limit theory more practical problem handling convenient, as well as in judging the convergence of the series, proves the function properties, function derivative problems in application, this is the article discusses problem, this research at home and abroad relatively less. Based on the existing Heine theorem several definitions, this paper on the basis of previous studies, combined with mathematical analysis in the related definition and Properties Analysis of Heine theorem, the connotation and the inference and proof of the corresponding, respectively, are given for different forms of the Heine theorem and Heine theorem, made corresponding promotion, and by using Heine theorem and the limit of a sequence of characters, the limit of a function related to nature, and judge the function limit exists, gives a detailed example, through case analysis of Heine theorem in optimizing the ultimate judgment and operation and other aspects of the application, reflects the Heine

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