Continuity, End Behavior, and Limits … · Continuity, End Behavior, and Limits The graph of a...
Transcript of Continuity, End Behavior, and Limits … · Continuity, End Behavior, and Limits The graph of a...
![Page 1: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/1.jpg)
Continuity,EndBehavior,andLimitsUnit1Lesson3
![Page 2: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/2.jpg)
Continuity,EndBehavior,andLimits
Studentswillbeableto:
Interpretkeyfeaturesofgraphsandtablesintermsofthequantities,
andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionof
therelationship.KeyVocabulary:Discontinuity,
Alimit,EndBehavior
![Page 3: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/3.jpg)
Continuity,EndBehavior,andLimits
Thegraphofacontinuousfunction hasnobreaks,holes,orgaps.Youcantracethegraphofacontinuousfunctionwithoutliftingyourpencil.Oneconditionforafunction𝒇 𝒙 tobecontinuousat𝒙 = 𝒄 isthatthefunctionmustapproachauniquefunctionvalueas𝒙 -valuesapproach𝒄 fromtheleftandrightsides.
![Page 4: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/4.jpg)
Continuity,EndBehavior,andLimits
Theconceptofapproachingavaluewithoutnecessarilyeverreachingitiscalledalimit.Ifthevalueof𝒇 𝒙 approachesauniquevalue𝑳 as𝒙approaches𝒄 fromeachside,thenthelimitof𝒇 𝒙 as𝒙approaches𝒄 is 𝑳.
𝐥𝐢𝐦𝒙→𝒄
𝒇 𝒙 = 𝑳
![Page 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/5.jpg)
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:• InfinitediscontinuityAfunctionhasaninfinitediscontinuityat𝒙 = 𝒄, ifthefunctionvalueincreasesordecreasesindefinitelyas𝒙approaches 𝒄 fromtheleftandright.
![Page 6: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/6.jpg)
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:• JumpdiscontinuityAfunctionhasajumpdiscontinuityat𝒙 = 𝒄 ifthelimitsofthefunctionas𝒙 approaches𝒄 fromtheleftandrightexistbuthavetwodistinctvalues.
![Page 7: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/7.jpg)
Continuity,EndBehavior,andLimits
Functionsthatarenotcontinuousarediscontinuous.Graphsthatarediscontinuouscanexhibit:• Removablediscontinuity,alsocalledpointdiscontinuity
Functionhasaremovablediscontinuityifthefunctioniscontinuouseverywhereexceptforaholeat𝒙 = 𝒄..
![Page 8: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/8.jpg)
Continuity,EndBehavior,andLimits
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)
.
![Page 9: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/9.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒂𝒕𝒙 = 𝟏
-40 -30 -20 -10 10 20 30 40
-30
-20
-10
10
20
30
x
y
![Page 10: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/10.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒂𝒕𝒙 = 𝟏
-40 -30 -20 -10 10 20 30 40
-30
-20
-10
10
20
30
x
y 𝒇 𝟏 = 𝟑 ∗ 𝟏𝟐 + 𝟏 − 𝟕 = −𝟑
𝒇 𝟏 𝒆𝒙𝒊𝒔𝒕𝒔
![Page 11: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/11.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒂𝒕𝒙 = 𝟏
-40 -30 -20 -10 10 20 30 40
-30
-20
-10
10
20
30
x
y 𝒙 → 𝟏<𝒚 → −𝟑
𝒙 𝟎. 𝟗 𝟎. 𝟗𝟗 𝟎. 𝟗𝟗𝟗
𝒇 𝒙 −𝟑. 𝟔𝟕 −𝟑. 𝟎𝟔𝟗𝟕 −𝟑. 𝟎𝟎𝟔𝟗𝟗𝟕
![Page 12: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/12.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒂𝒕𝒙 = 𝟏
-40 -30 -20 -10 10 20 30 40
-30
-20
-10
10
20
30
x
y 𝒙 → 𝟏A𝒚 → −𝟑
𝒙 𝟏. 𝟏 𝟏. 𝟎𝟏 𝟏. 𝟎𝟎𝟏
𝒇 𝒙 −𝟐. 𝟐𝟕 −𝟐. 𝟗𝟐𝟗𝟕 −𝟐. 𝟗𝟗𝟐𝟗𝟗𝟕
![Page 13: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/13.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.a. 𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒂𝒕𝒙 = 𝟏
-40 -30 -20 -10 10 20 30 40
-30
-20
-10
10
20
30
x
y 𝒇 𝟏 = −𝟑𝒂𝒏𝒅𝒚 → −𝟑𝒇𝒓𝒐𝒎𝒃𝒐𝒕𝒉𝒔𝒊𝒅𝒆𝒐𝒇𝒙 = 𝟏
𝐥𝐢𝐦𝒙→𝟏
𝟑𝒙𝟐 + 𝒙 − 𝟕 = 𝒇 (𝟏)
𝒇 𝒙 = 𝟑𝒙𝟐 + 𝒙 − 𝟕𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔𝒂𝒕𝒙 = 𝟏
![Page 14: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/14.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. 𝒇 𝒙 =𝟐𝒙𝒙 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
![Page 15: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/15.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. 𝒇 𝒙 =𝟐𝒙𝒙 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
𝒇 𝟎 =𝟐 ∗ 𝟎𝟎 =
𝟎𝟎
𝑻𝒉𝒆𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒊𝒔𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅𝒂𝒕𝒙 = 𝟎
![Page 16: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/16.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. 𝒇 𝒙 =𝟐𝒙𝒙 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒙 → 𝟎<𝒚 → −𝟐𝒙 −𝟎. 𝟏 −𝟎. 𝟎𝟏 −𝟎. 𝟎𝟎𝟏
𝒇 𝒙 −𝟐 −𝟐 −𝟐
![Page 17: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/17.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. 𝒇 𝒙 =𝟐𝒙𝒙 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒙 → 𝟎A𝒚 → 𝟐
𝒙 𝟎. 𝟏 𝟎. 𝟎𝟏 𝟎. 𝟎𝟎𝟏
𝒇 𝒙 𝟐 𝟐 𝟐
![Page 18: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/18.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
b. 𝒇 𝒙 =𝟐𝒙𝒙 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
𝒇 𝒙 =𝟐𝒙𝒙
𝒉𝒂𝒔𝒋𝒖𝒎𝒑𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚𝒂𝒕𝒙 = 𝟎
𝒔𝒊𝒏𝒄𝒆𝒚𝒗𝒂𝒍𝒖𝒆𝒔𝒂𝒓𝒆𝟐𝒂𝒏𝒅− 𝟐𝒐𝒏𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆𝒔𝒊𝒅𝒆𝒔𝒐𝒇𝒙 = 𝟎
![Page 19: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/19.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
![Page 20: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/20.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒇 −𝟐 =−𝟐 𝟐 − 𝟒−𝟐 + 𝟐 =
𝟎𝟎
𝒇 𝒙 𝒊𝒔𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅𝒂𝒕𝒙 = −𝟐
𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐
𝒊𝒔𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔𝒂𝒕𝒙 = −𝟐
![Page 21: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/21.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒙 → −𝟐<𝒚 → −𝟒
𝒙 −𝟐. 𝟏 −𝟐. 𝟎𝟏 −𝟐. 𝟎𝟎𝟏
𝒇 𝒙 −𝟒. 𝟏 −𝟒. 𝟎𝟏 −𝟒. 𝟎𝟎𝟏
![Page 22: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/22.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒙 → −𝟐A𝒚 → −𝟒𝒙 −𝟏. 𝟗 −𝟏. 𝟗𝟗 −𝟏. 𝟗𝟗𝟗
𝒇 𝒙 −𝟑. 𝟗 −𝟑. 𝟗𝟗 −𝟑. 𝟗𝟗𝟗
![Page 23: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/23.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
c. 𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐 𝒂𝒕𝒙 = −𝟐
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
𝒇 𝒙 =𝒙𝟐 − 𝟒𝒙 + 𝟐
𝒉𝒂𝒔𝒑𝒐𝒊𝒏𝒕𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚𝒂𝒕𝒙 = −𝟐𝒔𝒊𝒏𝒄𝒆 𝒚𝒗𝒂𝒍𝒖𝒆𝒊𝒔 − 𝟒𝒐𝒏𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆𝒔𝒊𝒅𝒆𝒔𝒐𝒇𝒙 = −𝟐
![Page 24: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/24.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
![Page 25: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/25.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
𝒇 𝟎 =𝟏
𝟑 ∗ 𝟎𝟐 = ∞
𝒇 𝒙 𝒊𝒔𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅𝒂𝒕𝒙 = 𝟎
𝒇 𝒙 =𝟏𝟑𝒙𝟐
𝒊𝒔𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔𝒂𝒕𝒙 = 𝟎
![Page 26: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/26.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒙 → 𝟎<𝒚 → +∞
𝒙 −𝟎. 𝟏 −𝟎. 𝟎𝟏 −𝟎. 𝟎𝟎𝟏
𝒇 𝒙 𝟑𝟑. 𝟑𝟑 𝟑, 𝟑𝟑𝟑. 𝟑𝟑 𝟑𝟑𝟑, 𝟑𝟑𝟑. 𝟑𝟑
![Page 27: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/27.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒙 → 𝟎A𝒚 → +∞
𝒙 𝟎. 𝟏 𝟎. 𝟎𝟏 𝟎. 𝟎𝟎𝟏
𝒇 𝒙 𝟑𝟑. 𝟑𝟑 𝟑, 𝟑𝟑𝟑. 𝟑𝟑 𝟑𝟑𝟑, 𝟑𝟑𝟑. 𝟑𝟑
![Page 28: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/28.jpg)
Continuity,EndBehavior,andLimits
SampleProblem1:Determinewhethereachfunctioniscontinuousatthegivenx-values.Justifyusingthecontinuitytest.Ifdiscontinuous,identifythetypeofdiscontinuity.
d. 𝒇 𝒙 =𝟏𝟑𝒙𝟐 𝒂𝒕𝒙 = 𝟎
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y 𝒇 𝒙 =𝟏𝟑𝒙𝟐
𝒉𝒂𝒔𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒚𝒅𝒊𝒔𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒊𝒕𝒚𝒂𝒕𝒙 = 𝟎𝒔𝒊𝒏𝒄𝒆 𝒚𝒗𝒂𝒍𝒖𝒆𝒊𝒔 + ∞
𝒘𝒉𝒆𝒏𝒙 → 𝟎
![Page 29: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/29.jpg)
Continuity,EndBehavior,andLimits
IntermediateValueTheorem
If𝒇 𝒙 isacontinuousfunctionand𝒂 < 𝒃 andthereisavalue𝒏 suchthat𝒏 isbetween𝒇 𝒂and𝒇 𝒃 ,thenthereisanumber𝒄,suchthat𝒂 < 𝒄 < 𝒃 and𝒇 𝒄 = 𝒏
![Page 30: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/30.jpg)
Continuity,EndBehavior,andLimits
TheLocationPrinciple
If𝒇 𝒙 isacontinuousfunctionand𝒇 𝒂 and𝒇 𝒃 haveoppositesigns,thenthereexistsatleastonevalue𝒄,suchthat𝒂 < 𝒄 < 𝒃 and𝒇 𝒄 = 𝟎.
Thatis,thereisazerobetween𝒂 and𝒃.
![Page 31: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/31.jpg)
Continuity,EndBehavior,andLimits
SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.
a. 𝒇 𝒙 = 𝒙 − 𝟑 𝟐 − 𝟒[𝟎, 𝟔]
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
![Page 32: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/32.jpg)
Continuity,EndBehavior,andLimits
SampleProblem2:Determinebetweenwhichconsecutiveintegerstherealzerosoffunctionarelocatedonthegiveninterval.
a. 𝒇 𝒙 = 𝒙 − 𝟑 𝟐 − 𝟒[𝟎, 𝟔]
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
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
![Page 33: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/33.jpg)
Continuity,EndBehavior,andLimits
EndBehavior
Theendbehaviorofafunctiondescribeswhatthe𝒚 -valuesdoas 𝒙 becomesgreaterandgreater.
When𝒙 becomesgreaterandgreater,wesaythat𝒙approachesinfinity,andwewrite𝒙 → +∞.
When𝒙 becomesmoreandmorenegative,wesaythat𝒙 approachesnegativeinfinity,andwewrite𝒙 →−∞.
![Page 34: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/34.jpg)
Continuity,EndBehavior,andLimits
Thesamenotationcanalsobeusedwith𝒚 or𝒇 𝒙andwithrealnumbersinsteadofinfinity.
Left- EndBehavior(as𝒙 becomesmoreandmorenegative): 𝐥𝐢𝐦
𝒙→<Y𝒇 𝒙
Right- EndBehavior(as𝒙 becomesmoreandmorepositive): 𝐥𝐢𝐦
𝒙→AY𝒇 𝒙
The𝒇 𝒙 valuesmayapproachnegativeinfinity,positiveinfinity,oraspecificvalue.
![Page 35: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/35.jpg)
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. 𝒇 𝒙 = −𝟑𝒙𝟑 + 𝟔𝒙 − 𝟏
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-8-7-6-5-4-3-2-1
12345678
x
y
![Page 36: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/36.jpg)
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.a. 𝒇 𝒙 = −𝟑𝒙𝟑 + 𝟔𝒙 − 𝟏
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-8-7-6-5-4-3-2-1
12345678
x
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
![Page 37: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/37.jpg)
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.
b. 𝒇 𝒙 =𝟒𝒙 − 𝟓𝟒 − 𝒙
-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
![Page 38: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/38.jpg)
Continuity,EndBehavior,andLimits
SampleProblem3: Usethegraphofeachfunctiontodescribeitsendbehavior.Supporttheconjecturenumerically.
b. 𝒇 𝒙 =𝟒𝒙 − 𝟓𝟒 − 𝒙 Fromthegraph,itappearsthat:
𝒇 𝒙 → −𝟒𝒂𝒔𝒙 → −∞𝒇 𝒙 → −𝟒𝒂𝒔𝒙 → ∞Thetablesupportsthisconjecture.
-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
𝒙 −𝟏𝟎𝟒 −𝟏𝟎𝟑 𝟎 𝟏𝟎𝟑 𝟏𝟎𝟒
𝒚 −𝟑. 𝟗𝟗𝟖𝟗 −𝟑. 𝟗𝟖𝟗𝟎 −𝟏. 𝟐𝟓 −𝟒. 𝟎𝟎𝟏 −𝟒. 𝟎𝟎𝟏𝟏
![Page 39: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/39.jpg)
Continuity,EndBehavior,andLimits
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.
![Page 40: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/40.jpg)
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
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
![Page 41: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/41.jpg)
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
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y Fromthegraph,itappearsthat:Afunction𝒙𝟐 − 𝟑𝒙 + 𝟐 isdecreasingfor𝒙 < 𝟏. 𝟓Afunction𝒙𝟐 − 𝟑𝒙 + 𝟐 isincreasingfor𝒙 > 𝟏. 𝟓
![Page 42: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/42.jpg)
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
-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
![Page 43: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/43.jpg)
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
![Page 44: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/44.jpg)
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 Fromthegraph,itappearsthat:Afunction𝒙𝟑 − 𝟏
𝟐𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐
isincreasing:𝒙 < −𝟏. 𝟔𝟔𝒂𝒏𝒅𝒙 > 𝟐Afunction𝒙𝟑 − 𝟏
𝟐𝒙𝟐 − 𝟏𝟎𝒙 + 𝟐
isdecreasing:−𝟏. 𝟔𝟔 < 𝒙 < 𝟐
![Page 45: 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](https://reader030.fdocuments.in/reader030/viewer/2022041006/5eab92781431d863cb1b1b17/html5/thumbnails/45.jpg)
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 Thetablesupportsthisconjecture.
𝒙 −𝟐 −𝟏. 𝟔𝟔 −𝟏 𝟎 𝟐 𝟑
𝒚 𝟏𝟐 𝟏𝟐. 𝟔𝟓 𝟏𝟎. 𝟓 𝟐 −𝟏𝟐 −𝟓. 𝟓