Limits in calculus

LimitsandContinuityWorksheet

Showallworkonyourpaperasdescribedinclass.Videolinksareincludedthroughoutforinstructiononhowtodothevarioustypesofproblems.Important:Worktheproblemstomatcheverythingthatwasshowninthevideos.Forexample:Supposeavideoshows3waystodoaproblem,(suchasalgebraically,graphically,andnumerically),thenyourworkshouldshowthese3waysalso.Thatis,eachvideoisamodelfortheworkIwanttoseeonyourpaper.

ESSAY

1) The idea of a limit: Gotohttp://youtu.be/5vSUrN-nqwE andwatchandtakenotesonTheIdeaofaLimit.

2) DefinitionofLimit:Afunctionfisdefinedonanintervalaroundc,exceptperhapsatthepointx= c.Wedefinethelimitofthefunctionf(x)asxapproachesc,written lim

xcf(x),tobeanumberL(if

oneexists)suchthatf(x)isasclosetoLaswewantwheneverxissufficientlyclosetoc(butxc).IfLexists,wewrite lim

xcf(x)=L

3) Onesidedlimits:Gotohttp://tutorial.math.lamar.edu/Classes/CalcI/OneSidedLimits.aspx andtakenotesonRight-handedlimit,Left-handedlimit,Example1,3,andComputeproblemsa-linExample4.AnswerstoExample4arethereonthelink.

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethegraphtoevaluatethelimit.4) lim

x-1f(x)

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

1

-1

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y

1

-1

A) -1 B) 12

C) D) -12

1

5) limx0

f(x)

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y654321

-1-2-3-4-5-6

A) 1 B) doesnotexist C) -1 D) 0

6) limx0

f(x)

x-2 -1 1 2 3 4 5

y12

10

8

6

4

2

-2

-4

x-2 -1 1 2 3 4 5

y12

10

8

6

4

2

-2

-4

A) 0 B) 6 C) doesnotexist D) -1

2

7) limx0

f(x)

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

http://youtu.be/5Im6jcaoiAwA) 1 B) C) doesnotexist D) -1

8) limx0

f(x)

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

A) doesnotexist B) C) -1 D) 1

3

9) limx0

f(x)

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

A) -2 B) doesnotexist C) 0 D) 2

10) limx0

f(x)

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

A) 0 B) -2 C) 1 D) doesnotexist

4

11) limx0

f(x)

x-4 -3 -2 -1 1 2 3 4

y

4

3

2

1

-1

-2

-3

-4

x-4 -3 -2 -1 1 2 3 4

y

4

3

2

1

-1

-2

-3

-4

A) 2 B) -2 C) -1 D) doesnotexist

12) Find limx(-1)-

f(x)and limx(-1)+

f(x)

x-4 -2 2 4

y2

-2

-4

-6

x-4 -2 2 4

y2

-2

-4

-6

A) -5;-2 B) -7;-2 C) -7;-5 D) -2;-7

5

13) limx0

f(x)

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4

x-4 -3 -2 -1 1 2 3 4

y4

3

2

1

-1

-2

-3

-4


Solvetheproblem.14) Whatconditions,whenpresent,aresufficienttoconcludethatafunctionf(x)hasalimitasxapproachessome

valueofa?A) Thelimitoff(x)asxafromtheleftexists,thelimitof f(x) asxa fromtherightexists,andthesetwo

limitsarethesame.B) Eitherthelimitoff(x)asxafromtheleftexistsorthelimitof f(x) asxa fromtherightexistsC) Thelimitoff(x)asxafromtheleftexists,thelimitof f(x) asxa fromtherightexists,andatleastone

oftheselimitsisthesameasf(a).D) f(a)exists,thelimitoff(x)asxa fromtheleftexists,andthelimitof f(x) asxafromtherightexists.

Usethetableofvaluesofftoestimatethelimit.15) Letf(x)=x2+8x-2,find lim

x2f(x).http://youtu.be/YHiZPKZ_gfM

x 1.9 1.99 1.999 2.001 2.01 2.1f(x)

A)x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 16.692 17.592 17.689 17.710 17.808 18.789

;limit=17.70

B)x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 5.043 5.364 5.396 5.404 5.436 5.763

;limit=5.40

C)x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 16.810 17.880 17.988 18.012 18.120 19.210

;limit=18.0

D)x 1.9 1.99 1.999 2.001 2.01 2.1f(x) 5.043 5.364 5.396 5.404 5.436 5.763

;limit=

6

16) Letf(x)= x-4x-2

,find limx4

f(x).

x 3.9 3.99 3.999 4.001 4.01 4.1f(x)

A)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236;limit=5.10

B)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745;limit=1.20

C)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745;limit=

D)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485;limit=4.0

17) Letf(x)= x-4x2-5x+4

,find limx4

f(x).

x 3.9 3.99 3.999 4.001 4.01 4.1f(x)

A)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 0.2448 0.2344 0.2334 0.2332 0.2322 0.2226;limit=0.2333

B)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) -0.3448 -0.3344 -0.3334 -0.3332 -0.3322 -0.3226;limit=-0.3333

C)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 0.4448 0.4344 0.4334 0.4332 0.4322 0.4226;limit=0.4333

D)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 0.3448 0.3344 0.3334 0.3332 0.3322 0.3226;limit=0.3333

18) Letf(x)= sin(5x)x

,find limx0

f(x).

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x) 4.99791693 4.99791693

A) limitdoesnotexist B) limit= 0 C) limit= 4.5 D) limit= 5

7

19) Letf()=cos(5),find lim

0f().

x -0.1 -0.01 -0.001 0.001 0.01 0.1f() -8.7758256 8.7758256

A) limit=5 B) limitdoesnotexist C) limit= 0 D) limit= 8.7758256

Findthelimit.20) lim

x182

A) 18 B) 3 2 C) 2 D) 2

21) limx-9

(6x-10)

A) -64 B) 44 C) 64 D) -44

Giveanappropriateanswer.22) Let lim

x-3f(x)=1and lim

x-3g(x)=-10.Find lim

x-3[f(x)-g(x)]. http://youtu.be/E9fF0kbgShg

A) -9 B) -3 C) 11 D) 1

23) Let limx7

f(x)=4and limx7

g(x)=5.Find limx7

[f(x) g(x)].

A) 9 B) 5 C) 7 D) 20

24) Let limx-8

f(x)=-7and limx-8

g(x)=-4.Find limx-8

f(x)g(x)

.

A) 47

B) 74

C) -8 D) -3

Findthelimit.25) lim

x2(x3+5x2-7x+1)

A) 15 B) doesnotexist C) 0 D) 29

26) limx0

x3-6x+8x-2

A) 4 B) 0 C) -4 D) Doesnotexist

27) limx0

1+x-1x

http://youtu.be/tzUSjMBzBuk

A) 1/4 B) Doesnotexist C) 0 D) 1/2

8

Determinethelimitbysketchinganappropriategraph.

28) limx6-

f(x),wheref(x)= -2x-6 forx< 64x-5 forx6 http://youtu.be/MwTbTOgRSNg

For more info on graphing piecewise defined function in the calculator, seehttp://mathbits.com/mathbits/tisection/precalculus/piecewise.htm

A) -18 B) -4 C) -5 D) 19

29) limx6+

f(x),wheref(x)= -4x-3 forx< 65x-2 forx6

A) -27 B) 28 C) -2 D) -1

30) limx4+

f(x),wheref(x)= x2+4 forx40 forx=4

A) 0 B) 16 C) 12 D) 20

Findthelimit,ifitexists.

31) limx0

x3+12x2-5x

5x

A) 5 B) 0 C) Doesnotexist D) -1

32) limx10

x2-100x-10

A) 20 B) Doesnotexist C) 1 D) 10

33) limx-9

x2+17x+72x+9

http://youtu.be/G-sDRUmTbX0

A) 306 B) -1 C) 17 D) Doesnotexist

34) limx5

x2+3x-40x-5

A) 13 B) Doesnotexist C) 0 D) 3

35) limx3

x2-9x2-7x+12

A) -6 B) 0 C) - 3 D) Doesnotexist

36) limh0

(x+h)3-x3h

A) 3x2+3xh+h2 B) 3x2 C) Doesnotexist D) 0

37) limx9

9-x9-x

http://youtu.be/1sF82HMZz4o

A) 0 B) -1 C) Doesnotexist D) 1

9

Computethevaluesoff(x)andusethemtodeterminetheindicatedlimit.

38) Iff(x)=x3-6x+8x-2

,find limx0

f(x).

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x)

A)x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858;limit=

B)x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) -4.09476 -4.00995 -4.00100 -3.99900 -3.98995 -3.89526;limit=-4.0

C)x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) -2.18529 -2.10895 -2.10090 -2.99910 -2.09096 -2.00574;limit=-2.10

D)x -0.1 -0.01 -0.001 0.001 0.01 0.1

f(x) -1.22843 -1.20298 -1.20030 -1.19970 -1.19699 -1.16858;limit=-1.20

39) Iff(x)= x-4x-2

,find limx4

f(x).http://youtu.be/q0OgTtfx0uk

x 3.9 3.99 3.999 4.001 4.01 4.1f(x)

A)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 3.97484 3.99750 3.99975 4.00025 4.00250 4.02485;limit=4.0

B)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745;limit=1.20

C)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 1.19245 1.19925 1.19993 1.20007 1.20075 1.20745;limit=

D)x 3.9 3.99 3.999 4.001 4.01 4.1

f(x) 5.07736 5.09775 5.09978 5.10022 5.10225 5.12236;limit=5.10

10

Forthefunctionfwhosegraphisgiven,determinethelimit.40) Find lim

x-1-f(x)and lim

x-1+f(x).

x-6 -4 -2 2 4 6

y

4

2

-2

-4

-6

-8

-10

x-6 -4 -2 2 4 6

y

4

2

-2

-4

-6

-8

-10

A) -5;-2 B) -7;-5 C) -2;-7 D) -7;-2

41) Find limx2-

f(x)and limx2+

f(x).

x-4 -3 -2 -1 1 2 3 4

y

8

6

4

2

-2

-4

-6

-8

x-4 -3 -2 -1 1 2 3 4

y

8

6

4

2

-2

-4

-6

-8

A) doesnotexist;doesnotexist B) 1;1C) -4;3 D) 3;-4

11

42) Find limx2+

f(x).

x-2 -1 1 2 3 4 5 6 7

f(x)8

7

6

5

4

3

2

1

-1x-2 -1 1 2 3 4 5 6 7

f(x)8

7

6

5

4

3

2

1

-1

A) 5 B) 1.3 C) -1 D) 4

43) Find limx1-

f(x).

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

f(x)5

4

3

2

1

-1

-2

-3

-4

-5

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

f(x)5

4

3

2

1

-1

-2

-3

-4

-5

A) 2 B) 12

C) -1 D) doesnotexist

12

44) Find limx1+

f(x).

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

f(x)5

4

3

2

1

-1

-2

-3

-4

-5

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

f(x)5

4

3

2

1

-1

-2

-3

-4

-5

A) 3 12

B) doesnotexist C) 3 D) 4

45) Find limx0

f(x).

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

y5

4

3

2

1

-1

-2

-3

-4

-5

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

y5

4

3

2

1

-1

-2

-3

-4

-5


13

46) Find limx0

f(x).

x-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

y87654321

-1-2-3-4-5-6-7-8

x-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

y87654321

-1-2-3-4-5-6-7-8

A) 0 B) doesnotexist C) 2 D) -2

47) Find limx-1

f(x).

x-4 -2 2 4

y

4

2

-2

-4

A

x-4 -2 2 4

y

4

2

-2

-4

A

A) -1 B) -23

C) doesnotexist D) 23

14

48) Find limx

f(x).

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

f(x)5

4

3

2

1

-1

-2

-3

-4

-5

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

f(x)5

4

3

2

1

-1

-2

-3

-4

-5

A) -2 B) C) 0 D) doesnotexist

ESSAY

49)Writedownthisnote:Note:Thepreviousproblembringsupanissue:Wecansay lim

xaf(x)=indicatingthat

asxapproachesthevaluea,thef(x)valuesgetlargerwithoutbound.Butthelimitinthiscaseisstilltechnicallydoesnotexistsinceinfinityisnotanumber.Sothinkofitthisway:Infinityisjustaspecialwaythelimitisfailstoexistandwritinginfinitygivesusmoreinformation(asopposedtofailingbecausethelimitfromtheleftdoesnotequalthelimitfromtheright).


Findthelimit.

50) limx-2

1x+2

http://youtu.be/nTfty6Q4wNw

A) Doesnotexist B) - C) 1/2 D)

51) limx7+

1(x-7)2

A) - B) 0 C) D) -1

52) limx-4-

5x2-16

A) B) -1 C) 0 D) -

53) limx(/2)+

tanx http://youtu.be/VyGOMJ-O0l4

A) 0 B) - C) 1 D)

15

54) limx1-

x2-4x+3x3- x

A) -1 B) 0 C) D) -

55) limx4+

x2-6x+8x3-4x

A) 0 B) Doesnotexist C) D) -

56) limx3+

2x2-9

A) 0 B) - C) 1 D)

57) limx

7x-1http://youtu.be/110FiQsvSfI

A) -1 B) 1 C) -8 D) 6

58) limx-

55-(9/x2)

A) -54

B) - C) 1 D) 5

59) limx

x2-7x+9x3-6x2+14

A) 914

B) 0 C) 1 D)

60) limx-

-4x2-3x+6-18x2-4x+9

A) 29

B) C) 1 D) 23

61) limx-

cos4xx

A) 1 B) 0 C) 4 D) -

62) limx

9x3-5x2+3x-x3-2x+6

A) B) 32

C) -9 D) 9

16

63) limx

2x+115x-7

A) 215

B) 0 C) D) -17

ESSAY

64)DefinitionofContinuity:Gotohttp://www.youtube.com/watch?v=hlorAjS0xWE&feature=topics andwatchandtakenotesoneverything,andknowforatest.


Findallpointswherethefunctionisdiscontinuous.65)

A) x=4 B) None C) x= 2 D) x= 4,x= 2

66)

A) x=-2,x=1 B) x=1 C) None D) x= -2

67)

A) x=0,x=2 B) x=2 C) x= -2,x= 0,x= 2 D) x= -2,x= 0

68)

A) x=-2,x=6 B) x=6 C) None D) x= -2

17

69)

A) None B) x=1,x= 4,x= 5 C) x= 4 D) x= 1,x= 5

70)

A) x=1 B) x=0,x= 1 C) x= 0 D) None

71)

A) None B) x=3 C) x= 0 D) x= 0,x= 3

72)

A) None B) x=-2 C) x= 2 D) x= -2,x= 2

73)

A) x=-2,x=0,x=2 B) None C) x= 0 D) x= -2,x= 2

18

Provideanappropriateresponse.74) Isfcontinuousatf(1),thatis,atx=1?

f(x)=

-x2+1,4x,-2,-4x+84,

-1x

77) Isfcontinuouson(-2,4]?Note:Afunctioniscontinuousonanintervalifitiscontinuousateverypointintheinterval

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

d10

8

6

4

2

-2

-4

-6

-8

-10

(2, 0)

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

d10

8

6

4

2

-2

-4

-6

-8

-10

(2, 0)

A) Yes B) No

ESSAY

78) Go to http://www.youtube.com/watch?v=Lgr-1ZKPnR4 WatchandtakenotesontheIntermediateValuetheorem.UsetheIntermediateValueTheoremtoprovethat6x3+5x2+4x+7=0hasasolutionbetween-2and-1.

79) UsetheIntermediateValueTheoremtoprovethat2x4+10x3-6x-6=0hasasolutionbetween-5and-4.


Findnumbersaandb,ork,sothatfiscontinuousateverypoint.80)

f(x)=-7,ax+b,21,

x3

http://youtu.be/6d0WlFdk2QoA) a=4,b=33 B) a=-7,b= 21 C) a= 4,b= 9 D) Impossible

81)

f(x)=x2,ax+b,x+6,

x-2

A) a=7,b=-10 B) a=-7,b= 10 C) a= -7,b= -10 D) Impossible

82)

f(x)=3x+8,

kx+4,

ifx

ESSAY

83) FormalDefinitionofaLimit(alsocalledtheepsilon- deltadefinition):Gotohttp://www.youtube.com/watch?v=NAAhJ5ucYJg .WatchandtakenotesontheformaldefinitionoftheLimitandtheproofoftheproblemshowninthevideo .

84)AnotherlookattheFormalDefinitionofLimit:Gotohttp://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx andtakenotesontheformaldefinitionoftheLimitandtheexplanationandgraphfollowingthedefintion.


Solvetheproblem.85) Selectthecorrectstatementforthedefinitionofthelimit: lim

xx0f(x)= L

meansthat__________________A) ifgivenanynumber>0,thereexistsanumber > 0,suchthatforallx,

0

87)

x

y

0 x

y

0

y=5x-19.2

9

8.8

21.96 2.04

NOTTOSCALE

f(x)=5x-1x0=2L=9=0.2

http://youtu.be/oY-I0BD1Xg8A) 0.08 B) 0.4 C) 7 D) 0.04

88)

f(x)=-5x-1x0=-1L=4=0.2

x

y

0 x

y

0

y=-5x-14.2

4

3.8

-1-1.04 -0.96

NOTTOSCALE

A) 0.04 B) -0.04 C) 7 D) 0.4

22

89)

f(x)=-x+2x0=-2L=4=0.2

x

y

0 x

y

0

y=-x+24.2

4

3.8

-2.2 -2 -1.8

NOTTOSCALE

A) 0.4 B) -0.2 C) 6 D) 0.2

90)

x

y

0 x

y

0

y= x-2

1.251

0.75

2.5625 3 3.5625

NOTTOSCALE

f(x)= x-2x0=3L=1

=14

A) 2 B) 0.5625 C) 0.4375 D) 1

23

91)

x

y

0 x

y

0

y=2x2

3

2

1

10.71 1.22

NOTTOSCALE

f(x)=2x2x0=1L=2=1

A) 0.22 B) 0.51 C) 1 D) 0.29

92)

x

y

0 x

y

0

y=x2-19

8

7

32.83 3.16

NOTTOSCALE

f(x)=x2-1x0=3L=8=1

A) 0.17 B) 5 C) 0.16 D) 0.33

ESSAY

93) Watch and take notes on this video which shows what happens when you have the wrong limit andyou try to prove it correct vs. when you have the correct limit: http://youtu.be/5iFwvdkc3OU

24


Afunctionf(x),apointx0,thelimitoff(x)asxapproachesx0,andapositivenumberisgiven.Findanumber > 0

suchthatforallx,0


Useyourcalculatortoplotthefunctionnearthepointx0beingapproached.Fromyourplotguessthevalueofthelimit.Youcanalsouse2ndcalcvalueinthecalculatortoevaluatethefunctionatanxvalueverycloseto aontheleftandthenontherightandguessthelimitthatway.Fortheseexamples,.001awayfrom aiscloseenough.

103) limx25

x-5x-25

http://youtu.be/D6l-uv_Q56I

A) 15

B) 0 C) 110

D) 5

104) limx0

25-x-5x

A) - 110

B) 10 C) 110

D) 5

ESSAY

SolvetheproblemBYHANDwithALGEBRA

105) Evaluatelim

x5 x2-25x-5

.http://youtu.be/beP0b18Fktg

Solvetheproblem.

106) Evaluatelim

x4 x -4x-2

.http://youtu.be/beP0b18Fktg

(same as previous video)

107) Evaluate limx9

x-3x-9

.

108) Evaluate limx3

2x-6x2-4x+3

.

109) Evaluate limx3

x2-x-6x-3

.

110) Findthetrigonometriclimit: lim0

2cos

.Usecalculatormethods.Yourcalculatormustbeinradianmode,

notdegrees.

111) Findthetrigonometriclimit: lim0

tan

.Usecalculatormethods.Yourcalculatormustbeinradianmode,

notdegrees.

Note:Thefollowingproblemspertaintosums,differences,products,andquotientsoflimits.

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112) Gotohttp://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx#Extras_Limit_LimitProp Writedowntheninepropertiesoflimitslistedthere.Theproofsofeachpropertyarealsoshown.Knowtheproofsforproperties7,1,and2foratest.Theproofsareathttp://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx#Extras_Limit_LimitProp


Provideanappropriateresponse.113) Provideashortsentencethatsummarizesthegenerallimitprinciplegivenbytheformalnotation

limxa

[f(x)g(x)]= limxa

f(x) limxa

g(x)=LM,giventhat limxa

f(x)=Land limxa

g(x)=M.

A) Thelimitofasumoradifferenceisthesumorthedifferenceofthelimits.B) Thelimitofasumoradifferenceisthesumorthedifferenceofthefunctions.C) Thesumorthedifferenceoftwofunctionsiscontinuous.D) Thesumorthedifferenceoftwofunctionsisthesumoftwolimits.

114) Thestatementthelimitofaconstanttimesafunctionistheconstanttimesthelimitfollowsfromacombinationoftwofundamentallimitprinciples.Whatarethey?

A) Thelimitofaproductistheproductofthelimits,andaconstantiscontinuous.B) Thelimitofaconstantistheconstant,andthelimitofaproductistheproductofthelimits.C) Thelimitofafunctionisaconstanttimesalimit,andthelimitofaconstantistheconstant.D) Thelimitofaproductistheproductofthelimits,andthelimitofaquotientisthequotientofthelimits.

Giveanappropriateanswer.115) Let lim

x2f(x)=-5and lim

x2g(x)=-8.Find lim

x2[f(x)- g(x)].

A) -5 B) -13 C) 3 D) 2

116) Let limx-9

f(x)=-1and limx-9

g(x)= 10.Find limx-9

[f(x) g(x)].

A) -9 B) 9 C) 10 D) -10

117) Let limx-6

f(x)=9and limx-6

g(x)=8.Find limx-6

f(x)g(x)

.

A) 98

B) 1 C) -6 D) 89

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AnswerKeyTestname:LIMITSANDCONTINUITYWORKSHEETUPDATED

1)2)3)4) B5) D6) A7) C8) A9) A10) B11) B12) D13) B14) A15) C16) D17) D18) D19) B20) C21) A22) C23) D24) B25) A26) C27) D28) A29) B30) D31) D32) A33) B34) A35) A36) B37) C38) B39) A40) C41) D42) D43) A44) C45) B46) B47) D48) B49)50) A

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51) C52) A53) B54) A55) A56) D57) A58) C59) B60) A61) B62) C63) A64)65) A66) B67) C68) B69) A70) D71) B72) D73) C74) B75) A76) A77) B78) Letf(x)=6x3+5x2+4x+7andlety0=0.f(-2)=-29andf(-1)=2.Sincefiscontinuouson[-2,-1]andsincey0=0

isbetweenf(-2)andf(-1),bytheIntermediateValueTheorem,thereexistsacintheinterval(-2,-1)withthepropertythatf(c)=0.Suchacisasolutiontotheequation6x3+5x2+4x+7=0.

79) Letf(x)=2x4+10x3-6x-6andlety0=0.f(-5)=24andf(-4)=-110.Sincefiscontinuouson[-5,-4]andsincey0=0isbetweenf(-5)andf(-4),bytheIntermediateValueTheorem,thereexistsacintheinterval(-5,-4)withthepropertythatf(c)=0.Suchacisasolutiontotheequation2x4+10x3-6x-6=0.

80) C81) C82) A83)84)85) B86) C87) D88) A89) D90) C91) A92) C93)94) A95) A

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96) B97) A98) A99) C100) A101) D102)

Let>0begiven.Choose=/5.Then0

Limits in calculus

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