AP 微积分关于Limit极限的题型总结带答案.docx
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AP 微积分关于Limit极限的题型总结带答案.docx
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AP微积分关于Limit极限的题型总结带答案
1、
is
Tosolvethisproblem,youneedtorememberhowtoevaluatelimits.Alwaysdolimitproblemsonthefirstpass.Wheneverwehavealimitofapolynomialfractionwhere x→ ∞,wedividethenumeratorandthedenominator,separately,bythehighestpowerof x inthefraction.
Nowtakethelimit.Rememberthatthe
if n >0,where k isaconstant.Thus,weget
2、
Ifwetakethelimitas x goesto0,wegetanindeterminateform
,solet'suseL'Hôpital'sRule.Wetakethederivativeofthenumeratorandthedenominatorandweget:
Now,whenwetakethelimitweget:
.
3、
Therearetwoimportanttrigonometriclimitstomemorize.
Thefirststepthatwealwaystakewhenevaluatingthelimitofatrigonometricfunctionistorearrangethefunctionsothatitlookslikesomecombinationofthelimitsabove.Wecandothisbyfactoringsin x outofthenumerator.Nowwecanbreakthisintolimitsthatwecaneasilyevaluate.
Nowifwetakethelimitas x →0weget4
(1)(0)=0.
4、
Thismay appear tobealimitproblem,butitis actually testingtoseewhetheryouknowthedefinitionofthederivative.:
Youshouldrecallthatthedefinitionofthederivativesays
Thus,ifwereplace f (x)withtan(x),wecanrewritetheproblemas
Thederivativeoftan x issec2x.Thus,
5、
Wetakethelimitas x goesto0,wegetanindeterminateform
solet'suseL'Hôpital'sRule.Wetakethederivativeofthenumeratorandthedenominatorandweget
.
Whenwetakethelimit,weagaingetanindeterminateform
solet'suseL'Hôpital'sRuleasecondtime.Wetakethederivativeofthenumeratorandthedenominatorandweget
.
Now,whenwetakethelimitweget:
.
6、
Noticethatifweplug5intotheexpressionsinthenumeratorandthedenominator,weget
whichisundefined.Beforewegiveup,weneedtoseeifwecansimplifythelimitsothatitcanbeevaluated.Ifwefactortheexpressioninthenumerator,weget
whichcanbesimplifiedto x +5.Now,ifwetakethelimit(bypluggingin5for x),weget10.
Noticethatifweplug5intotheexpressionsinthenumeratorandthedenominator,weget
whichisundefined.Beforewegiveup,weneedtoseeifwecansimplifythelimitsothatitcanbeevaluated.Ifwefactortheexpressioninthenumerator,weget
whichcanbesimplifiedto x+5.Now,ifwetakethelimit(bypluggingin5for x),weget10.
7、 Evaluate
Noticehowthislimittakestheformofthedefinitionofthederivative,whichis
Here,ifwethinkof f(x)as5x4,thenthisexpressiongivesthederivativeof5x4 atthepoint x=
.
Thederivativeof5x4 is f′(x)=20x3.At x=
weget
.
8、 Find k sothat
iscontinuous forall x.
Inorderfor f(x)tobecontinuousatapoint c, therearethreeconditionsthatneedtobefulfilled.
(1) f(c)exists.
(2)
exists.(3)
First,let'scheckcondition
(1):
f (4)exists;it'sequalto k.Next,let'scheckcondition
(2).
Fromtheleftside,weget
Fromtherightside,weget
Therefore,thelimitexists,and
.
Now,let'scheckcondition(3).Inorderforthisconditiontobefulfilled, k mustequal8.
9、
Ifwetakethelimitas x goesto0,wegetanindeterminateform
solet'suseL'Hôpital'sRule.Wetakethederivativeofthenumeratorandthedenominatorandweget
.
Now,whenwetakethelimitweget:
.
10、
First,rewritethelimitas
Next,breakthefractioninto
Now,ifwemultiplythetopandbottomofthefirstfractionby8,weget
Now,wecantakethelimit,whichgivesus8
(1)
(1)=8.
11、
Ifwetakethelimitas x goesto0,wegetanindeterminateform
solet'suseL'Hôpital'sRule.Wetakethederivativeofthenumeratorandthedenominatorandweget
.
Whenwetakethelimit,weagaingetanindeterminateform
solet'suseL'Hôpital'sRuleasecondtime.Wetakethederivativeofthenumeratorandthedenominatorandweget .
Now,whenwetakethelimitweget
.
12、
Ifwetakethelimitas x goesto∞,wegetanindeterminateform
,solet'suseL'Hôpital'sRule.Wetakethederivativeofthenumeratorandthedenominatorandweget
.
Whenwetakethelimit,weagaingetanindeterminateform
solet'suseL'Hôpital'sRuleasecondtime.Wetakethederivativeofthenumeratorandthedenominatorandweget
13、
First,rewritethelimitas
Next,breaktheexpressionintotworationalexpressions.
Thiscanbebrokenupfurtherinto
Wewillevaluatethelimitofeachseparately.Firstexpression.Dividethetopandbottomby x:
Then,multiplythetopandbottomoftheupperexpressionby3,andthetopandbottomofthelowerexpressionby5 .Now,ifwetakethelimit,wegetSecondexpression
.
Thislimitisstraightforward:
.Thirdexpression
First,pulltheconstant,3,outofthelimit:
.Now,ifwemultiplythetopandbottomoftheexpressionby5,weget
Now,ifwetakethelimit,weget
.
Combinethethreenumbers,andweget
.
14、 Evaluate
.
Noticehowthislimittakestheformofthedefinitionofthederivative,whichis
Ifwethinkof f (x)assin x,thenthisexpressiongivesthederivativeofsin x atthepoint
.
Thederivativeofsin x is f′(x)=cos x.At
weget
15、
Usethedoubleangleformulaforsine,sin2θ=2sinθcosθ,torewritethelimitandthensolve:
16、
Ifwetakethelimitas x goesto
,wegetanindeterminateform
,solet'suseL'Hôpital'sRule.Wetakethedervativeofthenumeratorandthedenominatorandweget
.
Now,whenwetakethelimitweget
17、Whatis
?
Recallthedefinitionofthederivativesays:
andthederivativeofsec x istan x sec x.Thus,
.
Therefore,thelimitdoesnotexist.
18、
Ifwetakethelimitas x goesto0fromtheright,wegetanindeterminateform
solet'suseL'Hôpital'sRule.Wetakethederivativeofthenumeratorandthedenominatorandweget
.
Now,whenwetakethelimitweget
.
19、
EitheruseL'Hôpital'sruleorrecallthat
.
Inthiscase,
canberewrittenas
20、
21、Find
.
Whenyouinsertfor x,thelimitis
whichisindeterminate.First,rewritethelimitas
.Then,useL'Hôpital'sRuletoevaluatethelimit:
.Thislimitexistsandequals0.
22、
Noticethelimitisintheformofthedefinitionofthederivative.Youcouldevaluatethelimit,butifyouseethedefinitionofthederivativeandthemainfunction, f(x)=2x2,itiseasiertoevaluatethederivativedirectly.Thus,thesolutionis f′(x)=4x.
23、Find
.
UseL'Hôpital'sRule.
24、
Ifwetakethelimitas x goesto
fromtheleft,wegetanindeterminateform
solet'suseL'Hôpital'sRule.Wetakethederivativeofthenumeratorandthedenominatorandweget
Wecansimplifythisusingtrigidentities
Now,whenwetakethelimitweget
25、
UsetheRationalFunctionTheorem.
.
26、
27、
Since|x -2|=2- x if x <2,thelimitas
28、
Notethat
where f (x)=ln x.
29、
Nonexistent
30、
(where[x]isthegreatestintegerin x)is
Nonexistent,Here,
and
.
31、
Dividebothnumeratoranddenominatorby
32、
Thegivenlimitequalsf (x)=sin x.
33、
Nonexistent
34、
Sincethedegreesofnumeratoranddenominatorarethesame,thelimitas x→∞istheratioofthecoefficientsofthetermsofhighestdegree:
35、
isanindeterminateformofthetype
.Applying L’Hôpital’sRule,youhave
.
36、Thediagramaboveshowsthegraphofafunction ffor-2≤ x≤4.Whichofthefollowingstatementsis/aretrue?
I.
exists
II.
exists
III.
exists
A.IonlyB.IIonlyC.IandIIonlyD.I,II,andIII
Thecorrectansweris(C).
Examiningthegraph,notethat
.
Sincethetwoone-sidedlimitsareequal,
exists.StatementIistrue.Also,notethat
.
Therefore,statementIIistrue,butstatementIIIisfalsebecausethetwoone-sidedlimitsarenotthesame.
37、
Notethatas xapproaches-
.
.
.
.
38、
Notethatthedefinitionofthederivative
.
.
39、
.
Therefore,
isanindeterminateformof
.Applying L’Hôpital’sRule,youhave .
40、Whatisthe
,if
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