从洛必达法则谈起

XXXX数学与计算科学学院2011届毕业论文

n?0,1,2,???（2）limana?a?????an. ?S，求：lim12n??bn??b?b?????bn12n解：令pnk?bka，k?1,2,???,n，tn?n，n?0,1,2,???，则满足条件：

b1?b2?????bnbn（1）（2）bn?0知pnk?0；

?pnk??k?1nbk（3）由b1?b2?????bn?????1；

k?1b1?b2?????bnn发散，且bn?0知，limpnk?limn??bka（4）limtn?limn?S ?0；

n??b?b?????bn??n??b12nn 由托普里兹定理得：limn???pnktk?lim?k?1nbka?k

n??k?1b1?b2?????bnbkn ?lima1?a2?????an?S.

n??b?b?????b12n01nnCnx0?CnZx1?????CnZxn例4.3.3 设limxn?a，Z?0，求lim.

n??n??(1?Z)nkkkkCnZCnZk?1,2,???,n解：令pnk?，，则满足条件：（1）p??0； nknn(1?Z)(1?Z)kk01nnCnZCn?CnZ?????CnZ(1?Z)n（2）?pnk?????1； nnn(1?Z)(1?Z)k?0k?0(1?Z)nnn(n?1)???(n?k?1)CZn(n?1)???(n?k?1)kk1?2???n?Z?Z （3）pnk?(1?Z)n(1?Z)n(1?Z)nknknkk ?Z?0; n(1?Z)（4）limxn?a,

n??kk01nnCnZCnx0?CnZx1?????CnZxn由托普里兹定理得：：lim?pnkxk?x?lim?a nknn??n??(1?Z)(1?Z)k?0n

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XXXX数学与计算科学学院2011届毕业论文

结束语

在此论文中，首先从洛必达法则谈起，并给出用洛必达法则解决极限的类型，然后引出施笃兹定理，并加以证明以及介绍其相关应用，最后引出托普里兹定理，并加以证明以及介绍其相关应用.

参考文献

[1] 数学分析（第三版）华东师范大学数学系.高等教育出版社，2001 [2] 吉米多维奇BⅡ．数学分析习题集[M].李荣冻译.人民教育出版社，1958 [3] 费定珲.数学分析习题集题解[M].山东科技出版社,1980 [4] G.克莱鲍尔 数学分析[M].上海科学技术出版社, 1981

[5] 冯志敏,薛瑞.使用洛必达法则的实质及其注意事项[J]. 中国科技信息.2009，8 [6] 谢佳朋. 浅谈洛必达法则[J].科教纵横.2010,3

[7] 黄涛、申方. 施笃兹定理的相关问题及应用探讨[J].天中学刊.2008,10 [8]严于亚.托普里兹定理及应用[J].盐城职大电大学刊.1994,1

Talking from the L'Hospital Rule

Author: XXX Supervisor: XXX

Abstract L'Hospital Rule is to solve an important limit undetermined type and simple way. From the L'Hospital Rule talk about, and then leds to Stolz Theorem and the Toeplitz Theorem, this article is divided into three main parts to explore these content: The first part gives out the two types of L'Hospital Rule and the proof, and with the L'Hospital Rule to solve problems of various types of undetermined type limits; the second part gives out proof of Stolz Theorem and its applications; The third part gives out proof of Toeplitz Theorem and its application. Keywords L'Hospital Rule Stolz Theorem Toeplitz Theorem Cauchy Theorem

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