指数型的不定式极限

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1、指数型的不定式极限肖劲森广东石油化工学院理学院，茂名　525000摘要：指数型不定式极限是较为常见不定式极限，其极限存在时可能取任一实数。然而更多人关注的是在什么条件下其极限为1。本文将考虑00型、∞0型和1∞型不定式极限。首先,由自然对数函数的双边不等式，获得了00型、∞0型极限值为1的充分条件。在这个条件下，不需要考虑函数的可微性。进不步，再次利用自然对数函数双边不等式，建立了一种计算1∞型极限的简单方法。关键词：不定式极限，指数型，洛必达法则中图分类号：O13OntheindeterminateformsofexponentialtypeXiaoJin-SenDe

2、partmentofScience,GuangdongUniversityofPetrochemicalTechnology,Maoming525000,ChinaAbstract:Theindeterminateformsofexponentialtypeareverycommoninthelimitscomputationandtheirlimitsmaytakeanynumber.However,morepeopleareconcernedabouttheconditionsunderwhichthoselimitsare1.Nowthisarticleisdev

3、otedtothestudyofindeterminateformsofexponentialtypeincluding00,∞0and1∞.Firstofall,byatwo-sideinequalitiesforthenaturallogarithmfunction,weobtainamoregeneralsufficientconditionsfor00and∞0havelimit1.Undertheseconditions,thedifferentiabilityoffunctionsarenotrequired.What’smore,usingthatinequal

4、itieswegetasimplewaytocomputethelimitforthetype1∞.Keywords:indeterminateforms,exponentialtype,L’Hospital’sRule0IntroductionItisknowntousthattheindeterminateformsofexponentialtype00,∞0or1∞maytakeanyrealnumber.Fortheseindeterminateformslimf(x)g(x),onemaysetthepowerfunctiong(x)satisfylimg(

5、x)lnf(x)=αorg(x)=αlnf(x)(αisfiniteorinfinite),thenlimf(x)g(x)=eα.Thefollowingtwotypicalexamplesarerelatedto00:limxα/lnx=eα,x→0+lim(e−1/x)−αx=eα.x→0+基金项目：作者简介：肖劲森（1984-），男，讲师，调和分析.E-mail:jinsenxiao@yahoo.com。-1-Inmanybooksandpapers,theauthorsaremoreconcernedabout:underwhatconditionsdoest

6、hetype00(or∞0)havelimite0=1(see[1,2,3,4,5,6]).TheusualprocedureistotakethelogarithmandthenuseL’Hospital’sRule.Asufficientconditionfor00haslimit1,requiresthedifferentiabilityoffunctions,obtainedbyRotandoandKorn[2]:TheoremASupposethatfandgarenonzerorealanalyticfunctionsatx=0forwhichf(x)≥x→0x→

7、0x→0Anotherbeautifulresult,whichmakenouseofL’Hospital’sRule,isduetoBaxleyandHayashi[1]:x→0x→0x→0ForTheoremB,theauthorspointedoutthatα>0indicatesthetype00whileα<0indicates∞0.What’smore,nosmoothnessconditionsofthefunctionsarerequired.Nowthepurposeofthispaperistofurtherstudy