Chapter1LimitsandTheirPropertiesLimits

Theword“l(fā)imit”isusedineverydayconversationtodescribetheultimatebehaviorofsomething,asinthe“l(fā)imitofone’sendurance”orthe“l(fā)imitofone’spatience.”Inmathematics,theword“l(fā)imit”hasasimilarbutmoreprecisemeaning.

Supposeyoudrive200miles,andittakesyou4hours.Thenyouraveragespeedis:Ifyoulookatyourspeedometerduringthistrip,itmightread65mph.Thisisyourinstantaneousspeed.1.1RatesofChangeandLimitsArockfallsfromahighcliff.Thepositionoftherockisgivenby:After2seconds:averagespeed:Whatistheinstantaneousspeedat2seconds?1.1RatesofChangeandLimitsforsomeverysmallchangeintwhereh=someverysmallchangeintWecanusetheTI-84toevaluatethisexpressionforsmallerandsmallervaluesofh.1.1RatesofChangeandLimits180164.16.00164.016.000164.0016.0000164.0002Wecanseethatthevelocityapproaches64ft/secashbecomesverysmall.Wesaythatthevelocityhasalimitingvalueof64ashapproacheszero.(Notethathneveractuallybecomeszero.)1.1RatesofChangeandLimitsThelimitashapproacheszero:01.1RatesofChangeandLimitsDefinition:LimitLetcandLberealnumbers.Thefunction

fhaslimitLasxapproachesc

if,foranygivenpositivenumberε,thereisapositivenumberδsuchthatforallx,1.1RatesofChangeandLimitsaLfDNE=DoesNotExistafL1L21.1RatesofChangeandLimitsDefinition:OneSidedLimitsLeft-HandLimit:ThelimitoffasxapproachesafromtheleftequalsLisdenotedRight-HandLimit:ThelimitoffasxapproachesafromtherightequalsLisdenoted1.1RatesofChangeandLimits1.1RatesofChangeandLimitsDefinition:Limitifandonlyif

and1.1RatesofChangeandLimitsDNE=DoesNotExistPossibleLimitSituationsafaf1.1RatesofChangeandLimits123412Atx=1:lefthandlimitrighthandlimitvalueofthefunction

doesnotexistbecausetheleftandrighthandlimitsdonotmatch!1.1RatesofChangeandLimitsAtx=2:lefthandlimitrighthandlimitvalueofthefunctionbecausetheleftandrighthandlimitsmatch.1234121.1RatesofChangeandLimitsAtx=3:lefthandlimitrighthandlimitvalueofthefunctionbecausetheleftandrighthandlimitsmatch.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–FINDINGALIMITBYTABLES

Solution:Wemaketwotables,asshownbelow,onewithxapproaching3fromtheleft,andtheotherwithxapproaching3fromtheright.20Limits

IMPORTANT!Thistableshowswhatf(x)isdoingasxapproaches3.OrwehavethelimitofthefunctionasxapproachesWewritethisprocedurewiththefollowingnotation.x22.92.992.99933.0013.013.14f(x)89.89.989.998?10.00210.0210.212

Def:WewriteIfthefunctionalvalueoff(x)isclosetothesinglerealnumberLwheneverxiscloseto,butnotequalto,c.(oneithersideofc).

orasx→c,thenf(x)→L310HLimitsAsyouhavejustseenthegoodnewsisthatmanylimitscanbeevaluatedbydirectsubstitution.22LimitPropertiesTheserules,whichmaybeprovedfromthedefinitionoflimit,canbesummarizedasfollows. Forfunctionscomposedofaddition,subtraction,multiplication,division,powers,root,limitsmaybeevaluatedbydirectsubstitution,providedthattheresultingexpressionisdefined.Examples–FINDINGLIMITSBYDIRECTSUBSTITUTIONSubstitute4forx.Substitute6forx.Examples–FINDINGLIMITSBYDIRECTSUBSTITUTIONExample1FindExample2Find

Somealgebraicrulesoflimits1Example

Somealgebraicrulesoflimits2ExampleSomealgebraicrulesoflimits3ExampleExample3:Find

Example4Findifyoupluginsomeverysmallvaluesfor,youwillseethisfunctionapproaches.Anditdoes'ntmatterwhetherispositiveornegative,youstillget,lookatthegraphof

Thedenominatorispositiveinbothcases,

sothelimitisthesame.Example5

Becausetheright-handlimitisnotequaltotheleft-handlimit,thelimitdoesnotexist.Therearesomeveryimportantpointsthatweneedtoemphasizefromthelasttwoexamples.1)Iftheleft-handlimitofafunctionisnotequaltotheright-handlimitofthefunction,thenthelimitdoesnotexist.2)Alimitequaltoinfinityisnotthesameasalimitthatdoesnotexist,butsometimesyouwillseetheexpression"nolimit",whichservesbothpurposes.If,thelimit,technically,doesnotexist.3)Ifkisapositiveconstant,thenanddoesnotexist.4)Ifkisapositiveconstant,thenandExample6:Find

As

getsbiggerandbigger,thevalueofthefunctiongetssmallerandsmaller.Therefore,Example7:Find

It'sthesamesituationastheoneinExample6;asdecrease(approachesnegativeinfinity),thevalueofthefunctionincrease(approachesaero).Wewritehis,Somealgebraicrulesoflimits4Example8Find

Whenyouhavevariablesinboththetopandbottom,youcan'tjustplugintotheexpression.Youwillget.Wesolvethisbyusingthefollowingtechnique:Whenanexpressionconsistsofapolynomialsdividedbyanotherpolynomials,divideeachtermofthenumeratorandthedenominatorbythehighestpowerofthatappearsintheexpression.Thehighestpowerofinthiscaseis,sowedivideeverytermintheexpression(bothtopandbottom)by,likeso:Nowwhenwetalkthelimit,thetwotermscontainingapproachzero.We'releftwith.

Example9:FindDivideezchtermby.Youget:

Example10:FindDivideezchtermby.

Theotherpowersdon'tmatter,becausethey'reallgoingtodisappear.Nowwehavethreenewrulesforevaluatingthelimitofarationalexpressionasapproachesinfinity:1)Ifthehighestpowerofinarationalexpressionisinthenumerator,thenthelimitasapproachesinfinityisinfinity.Example:2)Ifthehighestpowerofinarationalexpressionisinthedenominator,thenthelimitasapproachesinfinityiszero.Example:3)Ifthehighestpowerofinarationalexpressionisthesameinboththenumeratoranddenominator,thenthelimitasapproachesinfinityisthecoefficientofthehighestterminthenumratordividedbythecoefficientofthehighestterminthedenomiator.Example:1.2LimitsoftrigonometricfunctionsRuleNo.1:Thismayseemstrange,butifyoulookatthegraphsoftheyhaveapproximatelythesameslopeneartheorigin(asgetsclosertozero).Sinceandthesineofareaboutthesameasapproacheszero,theirquotientwillbeveryclosetoone.Furthermore,because(reviewcosinevaluesifyoudon'tgetthis!),weknowthatNowwewillfindasecondrule.Let'sevaluatethelimitFirst，multiplythetopandbottomby.

Weget:

Nowsimplifythelimitto:Next,wecanusethetrigonometricidentityandrewritethelimitas:Now,breakthisintotwolimits:Thefirstlimitis-1（seeRuleNo.1)andthesecondis0,sothelimitis0.RuleNo.2:Example11:FindExample12:FindRuleNo.3:RuleNo.4:Example13:FindProblem1.FindProblem2.FindProblem3.FindProblem4.FindProblem5.FindProblem6.FindProblem7.FindTheorem1.2PropertiesofLimitsTheorem1.3LimitsofPolynomialandRationalFunctionsUseyourcalculatortodeterminethefollowing:(a)(b)1.2Limitsoftrigonometricfunctions1DNESupposethatcisaconstantandthefollowinglimitsexist2.1RatesofChangeandLimitsSupposethatcisaconstantandthefollowinglimitsexist2.1RatesofChangeandLimitswherenisapositiveinteger.wherenisapositiveinteger.wherenisapositiveinteger.wherenisapositiveinteger.2.1RatesofChangeandLimitsEvaluatethefollowinglimits.Justifyeachstepusingthelawsoflimits.16-5/4262.1RatesofChangeandLimitsIffisarationalfunctionorcomplex:Eliminatecommonfactors.Performlongdivision.Simplifythefunction(ifacomplexfraction)Ifradicalsexist,rationalizethenumeratorordenominator.Ifabsolutevaluesexist,useone-sidedlimitsandthefollowingproperty.2.1RatesofChangeandLimits3/2DNE1/2DNE2.1RatesofChangeandLimitsTheoremIff(x)g(x)whenxisneara(exceptpossiblyata)andthelimitsoffandgbothexistasxapproachesa,then

2.1RatesofChangeandLimitsTheSqueeze(Sandwich)TheoremIff(x)g(x)h(x)whenxisneara(exceptpossiblyata)andthen2.1RatesofChangeandLimitsShowthat:Themaximumvalueofsineis1,soTheminimumvalueofsineis-1,soSo:2.1RatesofChangeandLimitsBythesandwichtheorem:2.1RatesofChangeandLimits2.1RatesofChangeandLimitsTherefore,2.1RatesofChangeandLimitssimplifyanddividebysinθ2.1RatesofChangeandLimits2.1RatesofChangeandLimitsP(cos,sin)Q(1,0)Thenotationmeansthatthevaluesoff(x)canbemadearbitrarilylarge(aslargeasweplease)bytakingxsufficientlyclosetoa(oneitherside)butnotequaltoa.2.2LimitsInvolvingInfinityafVerticalAsymptote2.2LimitsInvolvingInfinityVerticalAsymptoteThelinex=aiscalledaverticalasymptoteofthecurvey=f(x)ifatleastoneofthefollowingstatementsistrue:2.2LimitsInvolvingInfinityf(x)=lnxhasaverticalasymptoteatx=0sincef(x)=tanxhasaverticalasymptoteatx=/2since2.2LimitsInvolvingInfinity2.2LimitsInvolvingInfinity-∞x=3x=1DeterminetheequationsoftheverticalasymptotesofFindthelimitLetfbeafunctiondefinedonsomeinterval(a,∞).Thenmeansthatthevalueoff(x)canbemadeasclosetoLaswelikebytakingxsufficientlylarge.2.2LimitsInvolvingInfinityHorizontalAsymptoteLf2.2LimitsInvolvingInfinity2.2LimitsInvolvingInfinityDefinitionEndBehaviorModelSupposethatfisarationalfunctionasfollows:HorizontalAsymptoteTheliney=Liscalledahorizontalasymptoteofthecurvey=f(x)ifeitheror2.2LimitsInvolvingInfinityf(x)=exhasahorizontalasymptoteaty=0since2.2LimitsInvolvingInfinityIfnisapositiveinteger,then2.2LimitsInvolvingInfinityFindthelimit 2.2LimitsInvolvingInfinity-1/32/31/3Findthelimit 2.2LimitsInvolvingInfinityUsesqueezetheorem2.2LimitsInvolvingInfinityAfunctioniscontinuousatapointifthelimitisthesameasthevalueofthefunction.Thisfunctionhasdiscontinuitiesatx=1andx=2.Itiscontinuousatx=0andx=4,becausetheone-sidedlimitsmatchthevalueofthefunction1234122.3ContinuityDefinition:ContinuityAfunctioniscontinuousatanumberaifThatis,1. f(a)isdefined2. exists3. 2.3ContinuityDefinition:OneSidedContinuityAfunctionfiscontinuousfromtherightatanumberaifandfiscontinuousfromtheleftataif2.3Continuity1.Removablediscontinuity2.3Continuity2.Infinitediscontinuity2.3Continuity3.Jumpdiscontinuity2.3Continuity4.Oscillatingdiscontinuity2.3ContinuityDefinition:ContinuityOnAnIntervalAfunctionfiscontinuousonanintervalifitiscontinuousateverynumberintheinterval.(Iffisdefinedononesideofanendpointoftheinterval,weunderstandcontinuousattheendpointstomeancontinuousfromtherightorcontinuousfromtheleft).2.3ContinuityTheorem

f+g

f–g

cf

fg

f/gifg(a)0

f(g(x))Iffandgarecontinuousataandcisaconstant,thenthefollowingfunctionsarealsocontinuousata:2.3ContinuityTheoremAnypolynomialiscontinuouseverywhere;thatis,itiscontinuouson=(-∞,∞).Anyrationalfunctioniscontinuouswheneveritisdefined;thatis,itiscontinuousonitsdomain.2.3ContinuityAnyofthefollowingtypesoffunctionsarecontinuousateverynumberintheirdomain:Polynomials;RationalFunctions,RootFunctions;TrigonometricFunctions;InverseTrigonometricFunctions;ExponentialFunctions;andLogarithmicFunctions.2.3ContinuityIffiscontinuousatband ,then .Inotherwords,2.3ContinuityIfgiscontinuousataandfiscontinuousatg(a),thenthecompositefunctionf(g(x))iscontinuousata.2.3ContinuityTheIntermediateValueTheoremSupposethatfiscontinuousontheclosedinterval[a,b]andletNbeanynumberbetweenf(a)andf(b).Thenthereexistsanumbercin(a,b)suchthatf(c)=N.afbf(a)f(b)cf(c)=N2.3ContinuityUsetheIntermediateValueTheoremtoshowthatthereisarootofthegivenequationinthespecifiedinterval.2.3ContinuityGraphContinuousatx=0?

GraphContinuousatx=0?00yesundefined0noundefinedDNEnoundefined1no00yesundefined1noundefinedDNEno0DNEnoundefined0noDefinition:LimitLetcandLberealnumbers.Thefunction

fhaslimitLasxapproachesc

if,foranygivenpositivenumberε,thereisapositivenumberδsuchthatforallx,2.3ContinuitySolutionSetc=1andf(x)=5x-3andL=2.Foranygiven>0,thereexistsa>0suchthat0<|x-1|<whenever|f(x)-2|<2.3Continuity|(5x-3)-2|<|5x-5|<5|x-1|<|x-1|</5Soif=/51-11+2+2-22.3ContinuitySolutionSetc=2andf(