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Continuity,EndBehavior,andLimitsUnit1Lesson3
Continuity,EndBehavior,andLimits
Studentswillbeableto:
Interpretkeyfeaturesofgraphsandtablesintermsofthequantities,
andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionof
therelationship.KeyVocabulary:Discontinuity,
Alimit,EndBehavior
Continuity,EndBehavior,andLimits
Thegraphofacontinuousfunction hasnobreaks,holes,orgaps.Youcantracethegraphofacontinuousfunctionwithoutliftingyourpencil.Oneconditionforafunction๐ ๐ tobecontinuousat๐ = ๐ isthatthefunctionmustapproachauniquefunctionvalueas๐ -valuesapproach๐ fromtheleftandrightsides.
Continuity,EndBehavior,andLimits
Theconceptofapproachingavaluewithoutnecessarilyeverreachingitiscalledalimit.Ifthevalueof๐ ๐ approachesauniquevalue๐ณ as๐approaches๐ fromeachside,thenthelimitof๐ ๐ as๐approaches๐ is ๐ณ.
๐ฅ๐ข๐ฆ๐โ๐
๐ ๐ = ๐ณ
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:โข InfinitediscontinuityAfunctionhasaninfinitediscontinuityat๐ = ๐, ifthefunctionvalueincreasesordecreasesindefinitelyas๐approaches ๐ fromtheleftandright.
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:โข JumpdiscontinuityAfunctionhasajumpdiscontinuityat๐ = ๐ ifthelimitsofthefunctionas๐ approaches๐ fromtheleftandrightexistbuthavetwodistinctvalues.
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:โข Removablediscontinuity,alsocalledpointdiscontinuity
Functionhasaremovablediscontinuityifthefunctioniscontinuouseverywhereexceptforaholeat๐ = ๐..
Continuity,EndBehavior,andLimits
ContinuityTestAfunction๐ ๐ iscontinuous at๐ = ๐, ifitsatisfiesthefollowingconditions:1. ๐ ๐ isdefinedat c.๐(๐)exists.2. ๐ ๐ approachesthesamefunctionvaluetotheleft
andrightof ๐. ๐ฅ๐ข๐ฆ๐โ๐
๐ ๐ ๐๐๐๐๐๐
3. Thefunctionvaluethat ๐ ๐ approachesfromeachsideof ๐ is ๐ ๐ . ๐ฅ๐ข๐ฆ
๐โ๐๐ ๐ = ๐ (๐)
.
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. ๐ ๐ = ๐๐๐ + ๐ โ ๐๐๐๐ = ๐
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. ๐ ๐ = ๐๐๐ + ๐ โ ๐๐๐๐ = ๐
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x
y ๐ ๐ = ๐ โ ๐๐ + ๐ โ ๐ = โ๐
๐ ๐ ๐๐๐๐๐๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. ๐ ๐ = ๐๐๐ + ๐ โ ๐๐๐๐ = ๐
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x
y ๐ โ ๐<๐ โ โ๐
๐ ๐. ๐ ๐. ๐๐ ๐. ๐๐๐
๐ ๐ โ๐. ๐๐ โ๐. ๐๐๐๐ โ๐. ๐๐๐๐๐๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. ๐ ๐ = ๐๐๐ + ๐ โ ๐๐๐๐ = ๐
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x
y ๐ โ ๐A๐ โ โ๐
๐ ๐. ๐ ๐. ๐๐ ๐. ๐๐๐
๐ ๐ โ๐. ๐๐ โ๐. ๐๐๐๐ โ๐. ๐๐๐๐๐๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. ๐ ๐ = ๐๐๐ + ๐ โ ๐๐๐๐ = ๐
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y ๐ ๐ = โ๐๐๐๐ ๐ โ โ๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐ = ๐
๐ฅ๐ข๐ฆ๐โ๐
๐๐๐ + ๐ โ ๐ = ๐ (๐)
๐ ๐ = ๐๐๐ + ๐ โ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ = ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. ๐ ๐ =๐๐๐ ๐๐๐ = ๐
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. ๐ ๐ =๐๐๐ ๐๐๐ = ๐
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๐ ๐ =๐ โ ๐๐ =
๐๐
๐ป๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐ = ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. ๐ ๐ =๐๐๐ ๐๐๐ = ๐
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x
y ๐ โ ๐<๐ โ โ๐๐ โ๐. ๐ โ๐. ๐๐ โ๐. ๐๐๐
๐ ๐ โ๐ โ๐ โ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. ๐ ๐ =๐๐๐ ๐๐๐ = ๐
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x
y ๐ โ ๐A๐ โ ๐
๐ ๐. ๐ ๐. ๐๐ ๐. ๐๐๐
๐ ๐ ๐ ๐ ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. ๐ ๐ =๐๐๐ ๐๐๐ = ๐
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๐ ๐ =๐๐๐
๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ = ๐
๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐ = ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. ๐ ๐ =๐๐ โ ๐๐ + ๐ ๐๐๐ = โ๐
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. ๐ ๐ =๐๐ โ ๐๐ + ๐ ๐๐๐ = โ๐
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x
y ๐ โ๐ =โ๐ ๐ โ ๐โ๐ + ๐ =
๐๐
๐ ๐ ๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐ = โ๐
๐ ๐ =๐๐ โ ๐๐ + ๐
๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ = โ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. ๐ ๐ =๐๐ โ ๐๐ + ๐ ๐๐๐ = โ๐
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x
y ๐ โ โ๐<๐ โ โ๐
๐ โ๐. ๐ โ๐. ๐๐ โ๐. ๐๐๐
๐ ๐ โ๐. ๐ โ๐. ๐๐ โ๐. ๐๐๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. ๐ ๐ =๐๐ โ ๐๐ + ๐ ๐๐๐ = โ๐
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y ๐ โ โ๐A๐ โ โ๐๐ โ๐. ๐ โ๐. ๐๐ โ๐. ๐๐๐
๐ ๐ โ๐. ๐ โ๐. ๐๐ โ๐. ๐๐๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. ๐ ๐ =๐๐ โ ๐๐ + ๐ ๐๐๐ = โ๐
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๐ ๐ =๐๐ โ ๐๐ + ๐
๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ = โ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐ = โ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. ๐ ๐ =๐๐๐๐ ๐๐๐ = ๐
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. ๐ ๐ =๐๐๐๐ ๐๐๐ = ๐
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๐ ๐ =๐
๐ โ ๐๐ = โ
๐ ๐ ๐๐๐๐๐ ๐๐๐๐๐๐ ๐๐๐ = ๐
๐ ๐ =๐๐๐๐
๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ = ๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. ๐ ๐ =๐๐๐๐ ๐๐๐ = ๐
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x
y ๐ โ ๐<๐ โ +โ
๐ โ๐. ๐ โ๐. ๐๐ โ๐. ๐๐๐
๐ ๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐๐๐, ๐๐๐. ๐๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. ๐ ๐ =๐๐๐๐ ๐๐๐ = ๐
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y ๐ โ ๐A๐ โ +โ
๐ ๐. ๐ ๐. ๐๐ ๐. ๐๐๐
๐ ๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐๐๐, ๐๐๐. ๐๐
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. ๐ ๐ =๐๐๐๐ ๐๐๐ = ๐
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y ๐ ๐ =๐๐๐๐
๐๐๐๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ = ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ + โ
๐๐๐๐๐ โ ๐
Continuity,EndBehavior,andLimits
IntermediateValueTheorem
If๐ ๐ isacontinuousfunctionand๐ < ๐ andthereisavalue๐ suchthat๐ isbetween๐ ๐and๐ ๐ ,thenthereisanumber๐,suchthat๐ < ๐ < ๐ and๐ ๐ = ๐
Continuity,EndBehavior,andLimits
TheLocationPrinciple
If๐ ๐ isacontinuousfunctionand๐ ๐ and๐ ๐ haveoppositesigns,thenthereexistsatleastonevalue๐,suchthat๐ < ๐ < ๐ and๐ ๐ = ๐.
Thatis,thereisazerobetween๐ and๐.
Continuity,EndBehavior,andLimits
SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.
a. ๐ ๐ = ๐ โ ๐ ๐ โ ๐[๐, ๐]
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Continuity,EndBehavior,andLimits
SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.
a. ๐ ๐ = ๐ โ ๐ ๐ โ ๐[๐, ๐]
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๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐
๐ ๐ ๐ โ๐ โ๐ โ๐ ๐ ๐
๐ ๐ ispositiveand๐ ๐ isnegative,๐ ๐ ๐๐๐๐๐๐๐๐๐๐๐๐ ๐ โค ๐ โค ๐๐ ๐ isnegativeand๐ ๐ ispositive,๐ ๐ ๐๐๐๐๐๐๐๐๐๐๐๐๐ โค ๐ โค ๐๐ ๐ haszerosinintervals:๐ โค ๐ โค ๐๐๐๐ ๐ โค ๐ โค ๐
Continuity,EndBehavior,andLimits
EndBehavior
Theendbehaviorofafunctiondescribeswhatthe๐ -valuesdoas ๐ becomesgreaterandgreater.
When๐ becomesgreaterandgreater,wesaythat๐approachesinfinity,andwewrite๐ โ +โ.
When๐ becomesmoreandmorenegative,wesaythat๐ approachesnegativeinfinity,andwewrite๐ โโโ.
Continuity,EndBehavior,andLimits
Thesamenotationcanalsobeusedwith๐ or๐ ๐andwithrealnumbersinsteadofinfinity.
Left- EndBehavior(as๐ becomesmoreandmorenegative): ๐ฅ๐ข๐ฆ
๐โ<Y๐ ๐
Right- EndBehavior(as๐ becomesmoreandmorepositive): ๐ฅ๐ข๐ฆ
๐โAY๐ ๐
The๐ ๐ valuesmayapproachnegativeinfinity,positiveinfinity,oraspecificvalue.
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. ๐ ๐ = โ๐๐๐ + ๐๐ โ ๐
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Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. ๐ ๐ = โ๐๐๐ + ๐๐ โ ๐
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y Fromthegraph,itappearsthat:
๐ ๐ โ โ๐๐๐ โ โโ๐ ๐ โ โโ๐๐๐ โ โThetablesupportsthisconjecture.
๐ โ๐๐๐ โ๐๐๐ ๐ ๐๐๐ ๐๐๐
๐ ๐ โ ๐๐๐๐ ๐ โ ๐๐๐ โ๐ -๐ โ ๐๐๐ -๐ โ ๐๐๐๐
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.
b. ๐ ๐ =๐๐ โ ๐๐ โ ๐
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Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.
b. ๐ ๐ =๐๐ โ ๐๐ โ ๐ Fromthegraph,itappearsthat:
๐ ๐ โ โ๐๐๐๐ โ โโ๐ ๐ โ โ๐๐๐๐ โ โThetablesupportsthisconjecture.
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๐ โ๐๐๐ โ๐๐๐ ๐ ๐๐๐ ๐๐๐
๐ โ๐. ๐๐๐๐ โ๐. ๐๐๐๐ โ๐. ๐๐ โ๐. ๐๐๐ โ๐. ๐๐๐๐
Continuity,EndBehavior,andLimits
Increasing,Decreasing,andConstantFunctionsAfunction๐ isincreasingonaninterval๐ฐ ifandonlyifforevery๐ and๐containedin๐ฐ, ๐ < ๐ ๐ ,whenever๐ < ๐ .
A function ๐ is decreasing on an interval ๐ฐ if and only if for every ๐ and ๐contained in ๐ฐ, ๐ ๐ > ๐ ๐ whenever ๐ < ๐ .
A function ๐ remains constant on an interval ๐ฐ if and only if for every ๐and ๐ contained in ๐ฐ, ๐ ๐ = ๐ ๐ whenever ๐ < ๐ .
Points in the domain of a function where the function changes fromincreasing to decreasing or from decreasing to increasing are calledcritical points.
Continuity,EndBehavior,andLimitsSample Problem 4: Determine the interval(s) on which the function isincreasing and the interval(s) on which the function is decreasing.a. ๐ ๐ = ๐๐ โ ๐๐ + ๐
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Continuity,EndBehavior,andLimitsSample Problem 4: Determine the interval(s) on which the function isincreasing and the interval(s) on which the function is decreasing.a. ๐ ๐ = ๐๐ โ ๐๐ + ๐
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y Fromthegraph,itappearsthat:Afunction๐๐ โ ๐๐ + ๐ isdecreasingfor๐ < ๐. ๐Afunction๐๐ โ ๐๐ + ๐ isincreasingfor๐ > ๐. ๐
Continuity,EndBehavior,andLimitsSample Problem 4: Determine the interval(s) on which the function isincreasing and the interval(s) on which the function is decreasing.a. ๐ ๐ = ๐๐ โ ๐๐ + ๐
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Thetablesupportsthisconjecture.
๐ โ๐ ๐ ๐ ๐. ๐ ๐ ๐
๐ ๐ ๐ ๐ โ๐. ๐๐ โ๐. ๐ ๐
Continuity,EndBehavior,andLimitsSample Problem 4: Determine the interval(s) on which the function isincreasing and the interval(s) on which the function is decreasing.
b. ๐ ๐ = ๐๐ โ๐๐๐
๐ โ ๐๐๐ + ๐
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Continuity,EndBehavior,andLimitsSample Problem 4: Determine the interval(s) on which the function isincreasing and the interval(s) on which the function is decreasing.
b. ๐ ๐ = ๐๐ โ๐๐๐
๐ โ ๐๐๐ + ๐
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-8
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-2
2
4
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8
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12
x
y Fromthegraph,itappearsthat:Afunction๐๐ โ ๐
๐๐๐ โ ๐๐๐ + ๐
isincreasing:๐ < โ๐. ๐๐๐๐๐ ๐ > ๐Afunction๐๐ โ ๐
๐๐๐ โ ๐๐๐ + ๐
isdecreasing:โ๐. ๐๐ < ๐ < ๐
Continuity,EndBehavior,andLimitsSample Problem 4: Determine the interval(s) on which the function isincreasing and the interval(s) on which the function is decreasing.
b. ๐ ๐ = ๐๐ โ๐๐๐
๐ โ ๐๐๐ + ๐
-14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14
-12
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2
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x
y Thetablesupportsthisconjecture.
๐ โ๐ โ๐. ๐๐ โ๐ ๐ ๐ ๐
๐ ๐๐ ๐๐. ๐๐ ๐๐. ๐ ๐ โ๐๐ โ๐. ๐