Calculus Chapter 1 limits and its properties 极限及其性

Chapter1LimitsandTheirPropertiesLimitsTheword“limit”isusedineverydayconversationtodescribetheultimatebehaviorofsomething,asinthe“limitofone’sendurance”orthe“limitofone’spatience.”Inmathematics,theword“limit”hasasimilarbutmoreprecisemeaning.Supposeyoudrive200miles,andittakesyou4hours.Thenyouraveragespeedis:mi200mi4hr50hrdistanceaveragespeedelapsedtimextIfyoulookatyourspeedometerduringthistrip,itmightread65mph.Thisisyourinstantaneousspeed.1.1RatesofChangeandLimitsArockfallsfromahighcliff.Thepositionoftherockisgivenby:216ytAfter2seconds:216264yaveragespeed:av64ftft322secsecVWhatistheinstantaneousspeedat2seconds?1.1RatesofChangeandLimitsinstantaneousyVtforsomeverysmallchangeint22162162hhwhereh=someverysmallchangeintWecanusetheTI-84toevaluatethisexpressionforsmallerandsmallervaluesofh.1.1RatesofChangeandLimitsinstantaneousyVt22162162hhhyt1800.165.6.0164.16.00164.016.000164.0016.0000164.0002Wecanseethatthevelocityapproaches64ft/secashbecomesverysmall.Wesaythatthevelocityhasalimitingvalueof64ashapproacheszero.(Notethathneveractuallybecomeszero.)1.1RatesofChangeandLimits2016264limhhhThelimitashapproacheszero:20164464limhhhh2064641664limhhhh0lim6416hh0641.1RatesofChangeandLimitsDefinition:LimitLetcandLberealnumbers.ThefunctionfhaslimitLasxapproachescif,foranygivenpositivenumberε,thereisapositivenumberδsuchthatforallx,Lxfcx)(limεL||f(x)δc||x01.1RatesofChangeandLimitsaLfLxfax)(limDNExfax)(limDNE=DoesNotExistafL1L21.1RatesofChangeandLimitsDefinition:OneSidedLimitsLeft-HandLimit:ThelimitoffasxapproachesafromtheleftequalsLisdenotedLxfax)(limRight-HandLimit:ThelimitoffasxapproachesafromtherightequalsLisdenotedLxfax)(lim1.1RatesofChangeandLimits1.1RatesofChangeandLimitsDefinition:LimitLxfax)(limifandonlyifLxfax)(limLxfax)(limand1.1RatesofChangeandLimits)(limxfaxDNExfax)(limDNE=DoesNotExistPossibleLimitSituationsafaf1.1RatesofChangeandLimits123412Atx=1:1lim0xfx1lim1xfx11flefthandlimitrighthandlimitvalueofthefunction1limxfxdoesnotexistbecausetheleftandrighthandlimitsdonotmatch!1.1RatesofChangeandLimitsAtx=2:2lim1xfx2lim1xfx22flefthandlimitrighthandlimitvalueofthefunction2lim1xfxbecausetheleftandrighthandlimitsmatch.1234121.1RatesofChangeandLimitsAtx=3:3lim2xfx3lim2xfx32flefthandlimitrighthandlimitvalueofthefunction3lim2xfxbecausetheleftandrighthandlimitsmatch.1234121.1RatesofChangeandLimitsLimitsGivenafunctionf(x),ifxapproaching3causesthefunctiontotakevaluesapproaching(orequalling)someparticularnumber,suchas10,thenwewillcall10thelimitofthefunctionandwriteInpractice,thetwosimplestwayswecanapproach3arefromtheleftorfromtheright.LimitsForexample,thenumbers2.9,2.99,2.999,...approach3fromtheleft,whichwedenotebyx→3–,andthenumbers3.1,3.01,3.001,...approach3fromtheright,denotedbyx→3+.Suchlimitsarecalledone-sidedlimits.UsetablestofindExample1–FINDINGALIMITBYTABLESSolution:Wemaketwotables,asshownbelow,onewithxapproaching3fromtheleft,andtheotherwithxapproaching3fromtheright.20LimitsIMPORTANT!Thistableshowswhatf(x)isdoingasxapproaches3.OrwehavethelimitofthefunctionasxapproachesWewritethisprocedurewiththefollowingnotation.x22.92.992.99933.0013.013.14f(x)89.89.989.998?10.00210.0210.212x3lim2x410Def:WewriteIfthefunctionalvalueoff(x)isclosetothesinglerealnumberLwheneverxiscloseto,butnotequalto,c.(oneithersideofc).orasx→c,thenf(x)→L310Hxclimf(x)LLimitsAsyouhavejustseenthegoodnewsisthatmanylimitscanbeevaluatedbydirectsubstitution.22LimitPropertiesTheserules,whichmaybeprovedfromthedefinitionoflimit,canbesummarizedasfollows.Forfunctionscomposedofaddition,subtraction,multiplication,division,powers,root,limitsmaybeevaluatedbydirectsubstitution,providedthattheresultingexpressionisdefined.cxf(x))cf(limExamples–FINDINGLIMITSBYDIRECTSUBSTITUTIONx41.xlimSubstitute4forx.422x6x2.x3lim26364639Substitute6forx.Examples–FINDINGLIMITSBYDIRECTSUBSTITUTION•Example1Find•Example2Find25limxx33limxxSomealgebraicrulesoflimits1•Example•xfkxkfaxaxlimlim253limxxSomealgebraicrulesoflimits2•Example2121limlimlimthen,andfLLxgxfLxgLxfIaxaxax325limxxxSomealgebraicrulesoflimits3•Example2121limlimlimthen,andfLLxgxfLxgLxfIaxaxax1125limxxxExample3:Findxxx520limExample4Find•ifyoupluginsomeverysmallvaluesfor,youwillseethisfunctionapproaches.Anditdoes'ntmatterwhetherispositiveornegative,youstillget,lookatthegraphof201limxxx21xyxThedenominatorispositiveinbothcases,sothelimitisthesame.201limxx201limxx201limxxExample5•Becausetheright-handlimitisnotequaltotheleft-handlimit,thelimitdoesnotexist.xx1lim0-4-3-2-101234-4-3-2-11234xx1lim0xx1lim0Therearesomeveryimportantpointsthatweneedtoemphasizefromthelasttwoexamples.•1)Iftheleft-handlimitofafunctionisnotequaltotheright-handlimitofthefunction,thenthelimitdoesnotexist.•2)Alimitequaltoinfinityisnotthesameasalimitthatdoesnotexist,butsometimesyouwillseetheexpressionnolimit,whichservesbothpurposes.If,thelimit,te