极限的一个定理及其应用-外文翻译

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1、石家庄经济学院本科毕业设计（论文）外文翻译本科生毕业设计（论文）外文翻译学生姓名：张朋宇学号：408114010113专业班级：数学与应用数学指导教师：梁海燕老师2014年02月10日17石家庄经济学院本科毕业设计（论文）外文翻译ADiscussiononaLimitTheoremandItsApplicationAbstract:Thispaperproposesthatalimittheoremcanhelptosolveaspecificlimitproblemofsumformulaandthatsomelimitofpr

2、oductformulacanalsobesolvedbyexploitingthefeatureoflogarithmfunction.Keywords:limittheorem;sumformula;productformulaIncalculus,wewillusuallysolveaspecificlimitproblemofsumformulaButthissumformulacan’tsumdirectly,anditcan’tchangeintosomekindsoffunction’sintegralsum.Soi

3、tishardtoworkoutitslimit,forsolvethisproblem.Thispaper’sproposesisthatalimittheoremcanhelptosolvethislimitproblemofsumformulaandthatsomelimitofproductformulacanalsobesolvedbylogarithmfunction.Theorem1Let(a)fbedifferentiableatx=0andf(0)=0,(b)gbeintegrableforx∈[a,b].Weh

4、aveProof　Bythe(a),foreverythereisa>0suchthatimplies.Thenbythe(b),thereexistsarealnumberM>0suchthat

5、g(x)

6、≤Mforx∈17石家庄经济学院本科毕业设计（论文）外文翻译[a,b]andthereisa>0suchthat‖T‖

7、hecasethatf(0)≠0.Wecanshowthat17石家庄经济学院本科毕业设计（论文）外文翻译forf(0)>0andLetf(x)=xthentheorem1hasbecomeThisisdefinitionofdefiniteintegral,andbylogarithmfunctionwegetCorollary2Iffbedifferentiableatx=0andf(0)=1andgbeintegrableforxinto[a,b]thenwehaveInpracticalisusuallydivide[0,

8、1]intonparts,andchoose(k=1,2,…,n).Corollary3　Letfbedifferentiableatx=0andgbeintegrableforxinto[0,1],thenwehave(a)Iff(0)=0,wehave(b)(c)Iff(0)=1,wehave17石家庄经济学院本科毕业设计（论文）外文翻译Proof　Bythattheorem1andlogarithmfunction,wegetExample1　Evaluateeachofthefollowing:Solution　(a)Re

9、writethesumintheequivalentform17石家庄经济学院本科毕业设计（论文）外文翻译Sothatbytheorem1,（b）RewritethesumintheequivalentformSothatbytheorem1,Sothatbytheorem1,17石家庄经济学院本科毕业设计（论文）外文翻译（d）Letf(x)=sinaxandg(x)=x.ThenSothatbytheorem1,Sothatbytheorem1,Example2　Evaluatethefollowinglimits:Soluti

10、on(a)Wecanchangetheproductintoanequivalentfrombywriting17石家庄经济学院本科毕业设计（论文）外文翻译Letf(x)=1+xandg(x)=x.ThenSothatbycorollary2,(b