Continuity, End Behavior, and Limits … · Continuity, End Behavior, and Limits The graph of a continuous functionhas no breaks, holes, or gaps. You can trace the graph of a continuous

Continuity,EndBehavior,andLimitsUnit1Lesson3

Continuity,EndBehavior,andLimits

Studentswillbeableto:

Interpretkeyfeaturesofgraphsandtablesintermsofthequantities,

andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionof

therelationship.KeyVocabulary:Discontinuity,

Alimit,EndBehavior


Thegraphofacontinuousfunction hasnobreaks,holes,orgaps.Youcantracethegraphofacontinuousfunctionwithoutliftingyourpencil.Oneconditionforafunction𝒇 𝒙 tobecontinuousat𝒙 = 𝒄 isthatthefunctionmustapproachauniquefunctionvalueas𝒙 -valuesapproach𝒄 fromtheleftandrightsides.


Theconceptofapproachingavaluewithoutnecessarilyeverreachingitiscalledalimit.Ifthevalueof𝒇 𝒙 approachesauniquevalue𝑳 as𝒙approaches𝒄 fromeachside,thenthelimitof𝒇 𝒙 as𝒙approaches𝒄 is 𝑳.

𝐥𝐢𝐦𝒙→𝒄

𝒇 𝒙 = 𝑳


Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:• InfinitediscontinuityAfunctionhasaninfinitediscontinuityat𝒙 = 𝒄, ifthefunctionvalueincreasesordecreasesindefinitelyas𝒙approaches 𝒄 fromtheleftandright.


Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:• JumpdiscontinuityAfunctionhasajumpdiscontinuityat𝒙 = 𝒄 ifthelimitsofthefunctionas𝒙 approaches𝒄 fromtheleftandrightexistbuthavetwodistinctvalues.


Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:• Removablediscontinuity,alsocalledpointdiscontinuity

Functionhasaremovablediscontinuityifthefunctioniscontinuouseverywhereexceptforaholeat𝒙 = 𝒄..


ContinuityTestAfunction𝒇 𝒙 iscontinuous at𝒙 = 𝒄, ifitsatisfiesthefollowingconditions:1. 𝒇 𝒙 isdefinedat c.𝒇(𝒄)exists.2. 𝒇 𝒙 approachesthesamefunctionvaluetotheleft

andrightof 𝒄. 𝐥𝐢𝐦𝒙→𝒄

𝒇 𝒙 𝒆𝒙𝒊𝒔𝒕𝒔

3. Thefunctionvaluethat 𝒇 𝒙 approachesfromeachsideof 𝒄 is 𝒇 𝒄 . 𝐥𝐢𝐦

𝒙→𝒄𝒇 𝒙 = 𝒇 (𝒄)

.


SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒂𝒕𝒙 = 𝟏

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y 𝒇 𝟏 = 𝟑 ∗ 𝟏𝟐 + 𝟏 − 𝟕 = −𝟑

𝒇 𝟏 𝒆𝒙𝒊𝒔𝒕𝒔



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y 𝒙 → 𝟏<𝒚 → −𝟑

𝒙 𝟎. 𝟗 𝟎. 𝟗𝟗 𝟎. 𝟗𝟗𝟗

𝒇 𝒙 −𝟑. 𝟔𝟕 −𝟑. 𝟎𝟔𝟗𝟕 −𝟑. 𝟎𝟎𝟔𝟗𝟗𝟕



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x

y 𝒙 → 𝟏A𝒚 → −𝟑

𝒙 𝟏. 𝟏 𝟏. 𝟎𝟏 𝟏. 𝟎𝟎𝟏

𝒇 𝒙 −𝟐. 𝟐𝟕 −𝟐. 𝟗𝟐𝟗𝟕 −𝟐. 𝟗𝟗𝟐𝟗𝟗𝟕



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y 𝒇 𝟏 = −𝟑𝒂𝒏𝒅𝒚 → −𝟑𝒇𝒓𝒐𝒎𝒃𝒐𝒕𝒉𝒔𝒊𝒅𝒆𝒐𝒇𝒙 = 𝟏

𝐥𝐢𝐦𝒙→𝟏

𝟑𝒙𝟐 + 𝒙 − 𝟕 = 𝒇 (𝟏)

𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔𝒂𝒕𝒙 = 𝟏


SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.

b. 𝒇 𝒙 =𝟐𝒙𝒙 𝒂𝒕𝒙 = 𝟎

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𝒇 𝟎 =𝟐 ∗ 𝟎𝟎 =

𝟎𝟎

𝑻𝒉𝒆𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒊𝒔𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅𝒂𝒕𝒙 = 𝟎




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y 𝒙 → 𝟎<𝒚 → −𝟐𝒙 −𝟎. 𝟏 −𝟎. 𝟎𝟏 −𝟎. 𝟎𝟎𝟏

𝒇 𝒙 −𝟐 −𝟐 −𝟐




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y 𝒙 → 𝟎A𝒚 → 𝟐

𝒙 𝟎. 𝟏 𝟎. 𝟎𝟏 𝟎. 𝟎𝟎𝟏

𝒇 𝒙 𝟐 𝟐 𝟐




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𝒇 𝒙 =𝟐𝒙𝒙

𝒉𝒂𝒔𝒋𝒖𝒎𝒑𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚𝒂𝒕𝒙 = 𝟎

𝒔𝒊𝒏𝒄𝒆𝒚𝒗𝒂𝒍𝒖𝒆𝒔𝒂𝒓𝒆𝟐𝒂𝒏𝒅− 𝟐𝒐𝒏𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆𝒔𝒊𝒅𝒆𝒔𝒐𝒇𝒙 = 𝟎



c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐

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c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐

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y 𝒇 −𝟐 =−𝟐 𝟐 − 𝟒−𝟐 + 𝟐 =

𝟎𝟎

𝒇 𝒙 𝒊𝒔𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅𝒂𝒕𝒙 = −𝟐

𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐

𝒊𝒔𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔𝒂𝒕𝒙 = −𝟐



c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐

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y 𝒙 → −𝟐<𝒚 → −𝟒

𝒙 −𝟐. 𝟏 −𝟐. 𝟎𝟏 −𝟐. 𝟎𝟎𝟏

𝒇 𝒙 −𝟒. 𝟏 −𝟒. 𝟎𝟏 −𝟒. 𝟎𝟎𝟏



c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐

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y 𝒙 → −𝟐A𝒚 → −𝟒𝒙 −𝟏. 𝟗 −𝟏. 𝟗𝟗 −𝟏. 𝟗𝟗𝟗

𝒇 𝒙 −𝟑. 𝟗 −𝟑. 𝟗𝟗 −𝟑. 𝟗𝟗𝟗



c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐

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𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐

𝒉𝒂𝒔𝒑𝒐𝒊𝒏𝒕𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚𝒂𝒕𝒙 = −𝟐𝒔𝒊𝒏𝒄𝒆 𝒚𝒗𝒂𝒍𝒖𝒆𝒊𝒔 − 𝟒𝒐𝒏𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆𝒔𝒊𝒅𝒆𝒔𝒐𝒇𝒙 = −𝟐



d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎

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d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎

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𝒇 𝟎 =𝟏

𝟑 ∗ 𝟎𝟐 = ∞

𝒇 𝒙 𝒊𝒔𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅𝒂𝒕𝒙 = 𝟎

𝒇 𝒙 =𝟏𝟑𝒙𝟐

𝒊𝒔𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔𝒂𝒕𝒙 = 𝟎



d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎

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y 𝒙 → 𝟎<𝒚 → +∞

𝒙 −𝟎. 𝟏 −𝟎. 𝟎𝟏 −𝟎. 𝟎𝟎𝟏

𝒇 𝒙 𝟑𝟑. 𝟑𝟑 𝟑, 𝟑𝟑𝟑. 𝟑𝟑 𝟑𝟑𝟑, 𝟑𝟑𝟑. 𝟑𝟑



d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎

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y 𝒙 → 𝟎A𝒚 → +∞

𝒙 𝟎. 𝟏 𝟎. 𝟎𝟏 𝟎. 𝟎𝟎𝟏

𝒇 𝒙 𝟑𝟑. 𝟑𝟑 𝟑, 𝟑𝟑𝟑. 𝟑𝟑 𝟑𝟑𝟑, 𝟑𝟑𝟑. 𝟑𝟑



d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎

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y 𝒇 𝒙 =𝟏𝟑𝒙𝟐

𝒉𝒂𝒔𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒚𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚𝒂𝒕𝒙 = 𝟎𝒔𝒊𝒏𝒄𝒆 𝒚𝒗𝒂𝒍𝒖𝒆𝒊𝒔 + ∞

𝒘𝒉𝒆𝒏𝒙 → 𝟎


IntermediateValueTheorem

If𝒇 𝒙 isacontinuousfunctionand𝒂 < 𝒃 andthereisavalue𝒏 suchthat𝒏 isbetween𝒇 𝒂and𝒇 𝒃 ,thenthereisanumber𝒄,suchthat𝒂 < 𝒄 < 𝒃 and𝒇 𝒄 = 𝒏


TheLocationPrinciple

If𝒇 𝒙 isacontinuousfunctionand𝒇 𝒂 and𝒇 𝒃 haveoppositesigns,thenthereexistsatleastonevalue𝒄,suchthat𝒂 < 𝒄 < 𝒃 and𝒇 𝒄 = 𝟎.

Thatis,thereisazerobetween𝒂 and𝒃.


SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.

a. 𝒇 𝒙 = 𝒙 − 𝟑 𝟐 − 𝟒[𝟎, 𝟔]

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SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.

a. 𝒇 𝒙 = 𝒙 − 𝟑 𝟐 − 𝟒[𝟎, 𝟔]

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𝒙 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔

𝒚 𝟓 𝟎 −𝟑 −𝟒 −𝟑 𝟎 𝟓

𝒇 𝟎 ispositiveand𝒇 𝟐 isnegative,𝒇 𝒙 𝒄𝒉𝒂𝒏𝒈𝒆𝒔𝒊𝒈𝒏𝒊𝒏 𝟎 ≤ 𝒙 ≤ 𝟐𝒇 𝟒 isnegativeand𝒇 𝟔 ispositive,𝒇 𝒙 𝒄𝒉𝒂𝒏𝒈𝒆𝒔𝒊𝒈𝒏𝒊𝒏𝟒 ≤ 𝒙 ≤ 𝟔𝒇 𝒙 haszerosinintervals:𝟎 ≤ 𝒙 ≤ 𝟐𝒂𝒏𝒅𝟒 ≤ 𝒙 ≤ 𝟔


EndBehavior

Theendbehaviorofafunctiondescribeswhatthe𝒚 -valuesdoas 𝒙 becomesgreaterandgreater.

When𝒙 becomesgreaterandgreater,wesaythat𝒙approachesinfinity,andwewrite𝒙 → +∞.

When𝒙 becomesmoreandmorenegative,wesaythat𝒙 approachesnegativeinfinity,andwewrite𝒙 →−∞.


Thesamenotationcanalsobeusedwith𝒚 or𝒇 𝒙andwithrealnumbersinsteadofinfinity.

Left- EndBehavior(as𝒙 becomesmoreandmorenegative): 𝐥𝐢𝐦

𝒙→<Y𝒇 𝒙

Right- EndBehavior(as𝒙 becomesmoreandmorepositive): 𝐥𝐢𝐦

𝒙→AY𝒇 𝒙

The𝒇 𝒙 valuesmayapproachnegativeinfinity,positiveinfinity,oraspecificvalue.


SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. 𝒇 𝒙 = −𝟑𝒙𝟑 + 𝟔𝒙 − 𝟏

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SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. 𝒇 𝒙 = −𝟑𝒙𝟑 + 𝟔𝒙 − 𝟏

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y Fromthegraph,itappearsthat:

𝒇 𝒙 → ∞𝒂𝒔𝒙 → −∞𝒇 𝒙 → −∞𝒂𝒔𝒙 → ∞Thetablesupportsthisconjecture.

𝒙 −𝟏𝟎𝟒 −𝟏𝟎𝟑 𝟎 𝟏𝟎𝟑 𝟏𝟎𝟒

𝒚 𝟑 ∗ 𝟏𝟎𝟏𝟐 𝟑 ∗ 𝟏𝟎𝟗 −𝟏 -𝟑 ∗ 𝟏𝟎𝟗 -𝟑 ∗ 𝟏𝟎𝟏𝟐


SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.

b. 𝒇 𝒙 =𝟒𝒙 − 𝟓𝟒 − 𝒙

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SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.

b. 𝒇 𝒙 =𝟒𝒙 − 𝟓𝟒 − 𝒙 Fromthegraph,itappearsthat:

𝒇 𝒙 → −𝟒𝒂𝒔𝒙 → −∞𝒇 𝒙 → −𝟒𝒂𝒔𝒙 → ∞Thetablesupportsthisconjecture.

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𝒙 −𝟏𝟎𝟒 −𝟏𝟎𝟑 𝟎 𝟏𝟎𝟑 𝟏𝟎𝟒

𝒚 −𝟑. 𝟗𝟗𝟖𝟗 −𝟑. 𝟗𝟖𝟗𝟎 −𝟏. 𝟐𝟓 −𝟒. 𝟎𝟎𝟏 −𝟒. 𝟎𝟎𝟏𝟏


Increasing,Decreasing,andConstantFunctionsAfunction𝒇 isincreasingonaninterval𝑰 ifandonlyifforevery𝒂 and𝒃containedin𝑰, 𝒂 < 𝒇 𝒃 ,whenever𝒂 < 𝒃 .

A function 𝒇 is decreasing on an interval 𝑰 if and only if for every 𝒂 and 𝒃contained in 𝑰, 𝒇 𝒂 > 𝒇 𝒃 whenever 𝒂 < 𝒃 .

A function 𝒇 remains constant on an interval 𝑰 if and only if for every 𝒂and 𝒃 contained in 𝑰, 𝒇 𝒂 = 𝒇 𝒃 whenever 𝒂 < 𝒃 .

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. 𝒇 𝒙 = 𝒙𝟐 − 𝟑𝒙 + 𝟐

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x

y Fromthegraph,itappearsthat:Afunction𝒙𝟐 − 𝟑𝒙 + 𝟐 isdecreasingfor𝒙 < 𝟏. 𝟓Afunction𝒙𝟐 − 𝟑𝒙 + 𝟐 isincreasingfor𝒙 > 𝟏. 𝟓


-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

x

y

Thetablesupportsthisconjecture.

𝒙 −𝟏 𝟎 𝟏 𝟏. 𝟓 𝟐 𝟑

𝒚 𝟔 𝟐 𝟎 −𝟎. 𝟐𝟓 −𝟓. 𝟓 𝟐

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. 𝒇 𝒙 = 𝒙𝟑 −𝟏𝟐𝒙

𝟐 − 𝟏𝟎𝒙 + 𝟐

-14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

x

y


b. 𝒇 𝒙 = 𝒙𝟑 −𝟏𝟐𝒙

𝟐 − 𝟏𝟎𝒙 + 𝟐

-14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

x

y Fromthegraph,itappearsthat:Afunction𝒙𝟑 − 𝟏

𝟐𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐

isincreasing:𝒙 < −𝟏. 𝟔𝟔𝒂𝒏𝒅𝒙 > 𝟐Afunction𝒙𝟑 − 𝟏

𝟐𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐

isdecreasing:−𝟏. 𝟔𝟔 < 𝒙 < 𝟐


b. 𝒇 𝒙 = 𝒙𝟑 −𝟏𝟐𝒙

𝟐 − 𝟏𝟎𝒙 + 𝟐

-14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

x

y Thetablesupportsthisconjecture.

𝒙 −𝟐 −𝟏. 𝟔𝟔 −𝟏 𝟎 𝟐 𝟑

𝒚 𝟏𝟐 𝟏𝟐. 𝟔𝟓 𝟏𝟎. 𝟓 𝟐 −𝟏𝟐 −𝟓. 𝟓

Continuity, End Behavior, and Limits … · Continuity, End Behavior, and Limits The graph of a continuous functionhas no breaks, holes, or gaps. You can trace the graph of a continuous

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