Continuity,EndBehavior,andLimitsUnit1Lesson3
Continuity,EndBehavior,andLimits
Studentswillbeableto:
Interpretkeyfeaturesofgraphsandtablesintermsofthequantities,
andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionof
therelationship.KeyVocabulary:Discontinuity,
Alimit,EndBehavior
Continuity,EndBehavior,andLimits
Thegraphofacontinuousfunction hasnobreaks,holes,orgaps.Youcantracethegraphofacontinuousfunctionwithoutliftingyourpencil.Oneconditionforafunctionπ π tobecontinuousatπ = π isthatthefunctionmustapproachauniquefunctionvalueasπ -valuesapproachπ fromtheleftandrightsides.
Continuity,EndBehavior,andLimits
Theconceptofapproachingavaluewithoutnecessarilyeverreachingitiscalledalimit.Ifthevalueofπ π approachesauniquevalueπ³ asπapproachesπ fromeachside,thenthelimitofπ π asπapproachesπ is π³.
π₯π’π¦πβπ
π π = π³
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:β’ InfinitediscontinuityAfunctionhasaninfinitediscontinuityatπ = π, ifthefunctionvalueincreasesordecreasesindefinitelyasπapproaches π fromtheleftandright.
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:β’ JumpdiscontinuityAfunctionhasajumpdiscontinuityatπ = π ifthelimitsofthefunctionasπ approachesπ fromtheleftandrightexistbuthavetwodistinctvalues.
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:β’ Removablediscontinuity,alsocalledpointdiscontinuity
Functionhasaremovablediscontinuityifthefunctioniscontinuouseverywhereexceptforaholeatπ = π..
Continuity,EndBehavior,andLimits
ContinuityTestAfunctionπ π iscontinuous atπ = π, ifitsatisfiesthefollowingconditions:1. π π isdefinedat c.π(π)exists.2. π π approachesthesamefunctionvaluetotheleft
andrightof π. π₯π’π¦πβπ
π π ππππππ
3. Thefunctionvaluethat π π approachesfromeachsideof π is π π . π₯π’π¦
πβππ π = π (π)
.
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. π π = πππ + π β ππππ = π
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. π π = πππ + π β ππππ = π
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y π π = π β ππ + π β π = βπ
π π ππππππ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. π π = πππ + π β ππππ = π
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y π β π<π β βπ
π π. π π. ππ π. πππ
π π βπ. ππ βπ. ππππ βπ. ππππππ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. π π = πππ + π β ππππ = π
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y π β πAπ β βπ
π π. π π. ππ π. πππ
π π βπ. ππ βπ. ππππ βπ. ππππππ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. π π = πππ + π β ππππ = π
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y π π = βππππ π β βππππππππππππ ππππ = π
π₯π’π¦πβπ
πππ + π β π = π (π)
π π = πππ + π β ππππππππππππππππ = π
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. π π =πππ πππ = π
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. π π =πππ πππ = π
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π π =π β ππ =
ππ
π»πππππππππππππππ ππππππ πππ = π
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. π π =πππ πππ = π
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y π β π<π β βππ βπ. π βπ. ππ βπ. πππ
π π βπ βπ βπ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. π π =πππ πππ = π
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y π β πAπ β π
π π. π π. ππ π. πππ
π π π π π
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. π π =πππ πππ = π
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π π =πππ
ππππππππ πππππππππππππππ = π
πππππππππππππππππππ β ππππππππππππππ πππππ = π
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. π π =ππ β ππ + π πππ = βπ
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. π π =ππ β ππ + π πππ = βπ
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y π βπ =βπ π β πβπ + π =
ππ
π π πππππ ππππππ πππ = βπ
π π =ππ β ππ + π
πππ πππππππππππππππ = βπ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. π π =ππ β ππ + π πππ = βπ
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y π β βπ<π β βπ
π βπ. π βπ. ππ βπ. πππ
π π βπ. π βπ. ππ βπ. πππ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. π π =ππ β ππ + π πππ = βπ
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y π β βπAπ β βππ βπ. π βπ. ππ βπ. πππ
π π βπ. π βπ. ππ βπ. πππ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. π π =ππ β ππ + π πππ = βπ
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π π =ππ β ππ + π
πππππππππ πππππππππππππππ = βππππππ ππππππππ β ππππππππππππππ πππππ = βπ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. π π =ππππ πππ = π
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Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. π π =ππππ πππ = π
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π π =π
π β ππ = β
π π πππππ ππππππ πππ = π
π π =ππππ
πππ πππππππππππππππ = π
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. π π =ππππ πππ = π
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y π β π<π β +β
π βπ. π βπ. ππ βπ. πππ
π π ππ. ππ π, πππ. ππ πππ, πππ. ππ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. π π =ππππ πππ = π
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y π β πAπ β +β
π π. π π. ππ π. πππ
π π ππ. ππ π, πππ. ππ πππ, πππ. ππ
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. π π =ππππ πππ = π
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ππππππππππππ πππππππππππππππ = ππππππ ππππππππ + β
πππππ β π
Continuity,EndBehavior,andLimits
IntermediateValueTheorem
Ifπ π isacontinuousfunctionandπ < π andthereisavalueπ suchthatπ isbetweenπ πandπ π ,thenthereisanumberπ,suchthatπ < π < π andπ π = π
Continuity,EndBehavior,andLimits
TheLocationPrinciple
Ifπ π isacontinuousfunctionandπ π andπ π haveoppositesigns,thenthereexistsatleastonevalueπ,suchthatπ < π < π andπ π = π.
Thatis,thereisazerobetweenπ andπ.
Continuity,EndBehavior,andLimits
SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.
a. π π = π β π π β π[π, π]
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Continuity,EndBehavior,andLimits
SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.
a. π π = π β π π β π[π, π]
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π π π π π π π π
π π π βπ βπ βπ π π
π π ispositiveandπ π isnegative,π π ππππππππππππ π β€ π β€ ππ π isnegativeandπ π ispositive,π π πππππππππππππ β€ π β€ ππ π haszerosinintervals:π β€ π β€ ππππ π β€ π β€ π
Continuity,EndBehavior,andLimits
EndBehavior
Theendbehaviorofafunctiondescribeswhattheπ -valuesdoas π becomesgreaterandgreater.
Whenπ becomesgreaterandgreater,wesaythatπapproachesinfinity,andwewriteπ β +β.
Whenπ becomesmoreandmorenegative,wesaythatπ approachesnegativeinfinity,andwewriteπ βββ.
Continuity,EndBehavior,andLimits
Thesamenotationcanalsobeusedwithπ orπ πandwithrealnumbersinsteadofinfinity.
Left- EndBehavior(asπ becomesmoreandmorenegative): π₯π’π¦
πβ<Yπ π
Right- EndBehavior(asπ becomesmoreandmorepositive): π₯π’π¦
πβAYπ π
Theπ π valuesmayapproachnegativeinfinity,positiveinfinity,oraspecificvalue.
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. π π = βπππ + ππ β π
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Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. π π = βπππ + ππ β π
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π π β βπππ β ββπ π β ββπππ β βThetablesupportsthisconjecture.
π βπππ βπππ π πππ πππ
π π β ππππ π β πππ βπ -π β πππ -π β ππππ
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.
b. π π =ππ β ππ β π
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Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.
b. π π =ππ β ππ β π Fromthegraph,itappearsthat:
π π β βππππ β ββπ π β βππππ β βThetablesupportsthisconjecture.
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π βπππ βπππ π πππ πππ
π βπ. ππππ βπ. ππππ βπ. ππ βπ. πππ βπ. ππππ
Continuity,EndBehavior,andLimits
Increasing,Decreasing,andConstantFunctionsAfunctionπ isincreasingonanintervalπ° ifandonlyifforeveryπ andπcontainedinπ°, π < π π ,wheneverπ < π .
A function π is decreasing on an interval π° if and only if for every π and πcontained in π°, π π > π π whenever π < π .
A function π remains constant on an interval π° if and only if for every πand π contained in π°, π π = π π whenever π < π .
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. π π = ππ β ππ + π
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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. π π = ππ β ππ + π
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y Fromthegraph,itappearsthat:Afunctionππ β ππ + π isdecreasingforπ < π. πAfunctionππ β ππ + π isincreasingforπ > π. π
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. π π = ππ β ππ + π
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Thetablesupportsthisconjecture.
π βπ π π π. π π π
π π π π βπ. ππ βπ. π π
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. π π = ππ βπππ
π β πππ + π
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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. π π = ππ βπππ
π β πππ + π
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y Fromthegraph,itappearsthat:Afunctionππ β π
πππ β πππ + π
isincreasing:π < βπ. πππππ π > πAfunctionππ β π
πππ β πππ + π
isdecreasing:βπ. ππ < π < π
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. π π = ππ βπππ
π β πππ + π
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π βπ βπ. ππ βπ π π π
π ππ ππ. ππ ππ. π π βππ βπ. π