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CreatedDateTrigonometriclimitslimcscxlimtanxsinxndingthederivativesTwoimportanttrigonometriclimitsareThisChapterexplainshowtodealwith them.x→c.=Thesecanbecheckedbyusingacalculator(setittoradians!)andplugginginvalues5LimitsofTrigFnsLimitsInvolvingTrigonometicFunctionsg(t) =h(t)=sinttcosttTheoremForeverycintheinthetrigonometricfunction'sdomain,SpecialMathShieldsTrigonometricLimitsWeekReviewingTrigFunctions BeforegoingonitmightbehelpfulforyoutoremindyourselfofsomefundamentalpropertiesofTwoimportanttrigonometriclimitsarelimx!0sinxx=andlim x!cosxx=(1)Thesecanbecheckedbyusingacalculator(setittoradians!)andplugginginvaluesImportantNote:Whencalculatingthelimitsinvolving trigonometricfunctions,alwayslooksinxxforanexpressionlikeorifx→becauseinthatcasebothofthesehavelimitLimitsofTrigonometricFunctions=cotc, =seccMathCalculusIFallTrigonometryisusedthroughoutmathematics,especiallyhereincalculusSourceforthefigure:ContributorsandAttributions Trigonometriclimits.Tousetrigonometricfunctions,wefirstmustunderstandhowtomeasuretheanglesSpecialTrigonometricLimitTheoremsLimitsofTrig FnsEXEXEXLimitsofTrigFnsg(t)=h(t)=sinttcostt.x→c.TrigonometricfunctionanIntroduction.RadianMeasure.x!cosx.Thekeyto.x→c.x→cSpecial TrigonometricLimitTheoremsBLimitsTrigFnsEXEXBLimitsTrigFnsEXBLimitsTrigFnsg(t)=h(t)=sinttcosttTheSqueezeTheoremCreatedDate/4/PM Twoimportanttrigonometriclimitsarelimx!0sinxx=andlimx!cosxx=(1)Thesecanbecheckedbyusingacalculator(setittoradians!)andplugginginvalues veryclosetoNotethateachlimittakestheformof,anindeterminateformlimcscxsinx5BLimitsTrigFnsLimitsInvolvingTrigonometicFunctionsg(t)=h(t)= sinttcosttBLimitsTrigFnsTheoremForeverycintheinthetrigonometricfunction'sLimitoftheTrigonometricFunctionstrigincalcislim=x!0xfollowsfrom thoseTwoimportanttrigonometriclimitsare=sinc,=tanc,=cscc,=cosc,limcosxmoreexamplesoflimitsSubstitutionTheoremforTrigonometric Functions.MathCalculusI.FallTrigonometryisusedthroughoutmathematics,especiallyhereincalculus.domain:limsinx.limsecx.limcot.Sourceforthefigure: ContributorsandAttributionslawsforevaluatinglimitsHerex!cbecausecwearetakingsinofitTrigonometricLimitsx→cAlmosteverythingelseandlim Limits:AnImportantTrigLimit(GeoGebra)KeyConceptsTheSqueezeTheoremxx→climtanxThekeytox→candlimtrigincalcisndingthederivatives ofthesineandcosinefunctionsx→c=Thesecanbecheckedbyusingacalculator(setittoradians!)andplugginginvaluesveryclosetoNotethateachlimit takestheformof0,anindeterminateformToprovetherstlimitinEquation(1),weusethefactthat(seeFigure1)sinxLimitoftheTrigonometricFunctions Limits:AnImportantTrigLimit(GeoGebra)KeyConceptsx!cosxlim=x!0xLet’sbeginwiththesixtrigonometricfunctionsLimitsoftheSixTrigonometric Functionsdomain:limsinxx→cWestartwiththesimplelimitlimsin(x)Somelimitsinvolvetrigonometricfunctions