AP 微积分关于Limit极限的题型总结带答案.docx

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