極限理論在數(shù)學分析中地位與作用

1、1極限理論在數(shù)學分析中地位與作用摘要極限理論是數(shù)學分析的基本理論，極限概念是極限理論的核心。作為微積分的基礎，極限理論包括函數(shù)極限和數(shù)列極限。本文從連續(xù)、導數(shù)、定積分、以及級數(shù)的收斂性等方面求解極限，深入探索了極限問題所涉及的各個方面。首先從定義入手，找出函數(shù)極限與數(shù)列極限的聯(lián)系，進而運用極限的性質、判定準則、柯西極限理論、迫斂性等方法求解不同類型的函數(shù)、數(shù)列極限。在極限定義的基礎上提供了又一種求解極限的方法，即無窮小量替換法求解極限。

2、同時例舉了幾類特殊極限，對其求解計算，總結出一些重要規(guī)律及相關結論。這些結論奠定了極限理論在數(shù)學分析中的地位與作用，為后繼的學習與研究極限提供更好的判別方法和更完整的理論體系，對數(shù)學分析具有重大意義。關鍵詞：關鍵詞：極限；數(shù)列；函數(shù)；定積分；判定準則3目錄目錄1引言.......................................................................................

3、.........................................................42極限的思想淵源與發(fā)展史............................................................................................................52.1極限的思想及歷史..............................

4、..............................................................................52.1.1最早的無限分割思想..............................................................................................62.1.2.西方的窮竭法與中國的割圓術.............

5、................................................................62.1.3.斯杰文對窮竭法的簡化和瓦里斯的算術化.........................................................93極限的相關理論.................................................................

6、.........................................................103.1極限概念的逐步形成.......................................................................................................103.2極限概念的完善..................................

7、.............................................................................103.2.1函數(shù)極限.................................................................................................................123.2.2數(shù)列極限........

8、.................................................................................................................153.3極限理論的確立...............................................................................................

9、................163.3.1波爾查諾的工作.....................................................................................................163.3.2柯西的極限理論...........................................................................

10、.........................173.3.3維爾斯特拉斯的靜態(tài)理論....................................................................................174數(shù)學分析中極限的作用................................................................................

11、..............................174.1函數(shù)的連續(xù).......................................................................................................................194.2數(shù)列的收斂性..................................................

12、.................................................................214.2.1唯一性....................................................................................................................214.2.2有界性...................

13、...............................................................................................224.3導數(shù)是特殊的極限...........................................................................................................225極限

14、的計算..................................................................................................................................245.1利用導數(shù)的定義...........................................................................

15、....................................245.2利用初等函數(shù)的連續(xù)性...................................................................................................245.3數(shù)列極限.............................................................

16、..............................................................255.3.1利用函數(shù)極限求數(shù)列極限....................................................................................255.3.2利用定積分求數(shù)列極限.......................................

17、.................................................255.4函數(shù)極限...........................................................................................................................265.4.1利用迫斂性求函數(shù)極限......................

18、..................................................................265.4.2利用羅比達法則求函數(shù)極限.................................................................................265.4.3利用泰勒級數(shù)展開式求函數(shù)極限.................................

19、.......................................275.4.4利用中值定理求函數(shù)極限....................................................................................275.4.5利用定積分的定義求函數(shù)極限...........................................................

20、.................285.4.6利用等價無窮小替換求函數(shù)極限........................................................................285.4.7利用收斂級數(shù)的必要條件求函數(shù)極限................................................................285.5利用級數(shù)收斂的必要條件求極限....

21、................................................................................285.6.將數(shù)列的極限化為定積分................................................................................................296結論......................

22、........................................................................................................................307參考文獻.............................................................................................