limits in calculus
DESCRIPTION
limits and calculusTRANSCRIPT
-
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
A) 0 B) doesnotexist C) -3 D) 3
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
A) 1 B) doesnotexist C) -1 D) 0
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).
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
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.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
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.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
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.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
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
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MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Afunctionf(x),apointx0,thelimitoff(x)asxapproachesx0,andapositivenumberisgiven.Findanumber > 0
suchthatforallx,0
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MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
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
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
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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AnswerKeyTestname:LIMITSANDCONTINUITYWORKSHEETUPDATED
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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AnswerKeyTestname:LIMITSANDCONTINUITYWORKSHEETUPDATED
96) B97) A98) A99) C100) A101) D102)
Let>0begiven.Choose=/5.Then0