1、Bc“ 5 “ F“n Taa an=12null 0“b 2n F0“ b kknC=+=nknnknC02)11( (1) iK“AcB V 0“ (2) ! A B V “ 9 V “b AB 1 ! T BK“ 5 | Ta 1bT K“ Ai b T K“AiTa 212aa Ta 313aa 23aa b“V VK/ V “ nullnull ,21 naaaS = b TS 2 ! nullnull ,21 naaaA= nullnull ,21 nbbbB = 5 VV U ABAB nullnull ,2211 nnbababa= b / V p (1) 0 =(2) a ,
2、abc(3) , ab ,abc(4) , b ,ab ab = ,ab 1 “ b“b 0 0 2 “ V a ,abc a ,abc b 1 3 “ , 0“V ba, ,abc ba, ,abc b 4 “ , baba ba, ,ab, baba ,ab , baba ,ab b “|V U/ “ (1) xx+320 L 8 (2) BK 8 (3) v 0i Ol1 8 (4) Z L 8b 0cotsin =xx 1 b 32| yxyx O 3Qxxx AB AC= B C= (2) AB AC= B C= b |o pV UPH 5 ?wH 5b 1 ! cbaA ,= dc
3、bB ,= dcC ,= 5 AB AC= b CB 2 ! cbaA ,= edcB ,= dcC ,= 5 AB AC= b CB / 5 ? hb (1) BAx Ax i O Bx (2) BAx Ax Bx b 1 b BAx Ax Bx b 2 b BAx Ax i O Bx b 35 f 1. ! , =S ,T V ? ? t ? abc= , f ST V ? 2733= 6!3 = cbafnullnullnull: b bcafnullnullnull:acbfnullnullnull:cabfnullnullnull:bacfnullnullnull:abcfnulln
4、ullnull:2. (1) y uW , , WBB ab 01(2) y uW (, )01 (, )+ WBBb 1 1,0,: bafabaxyx=null 2 ),()1,0(: +f)cot()21tan( xxx =null b 3. |/ f f gf il (1) yfu= =() logau , u g x= =() x23 ; (2) yfu= =() arcsinu , u g x= =() ex (3) yfu= =() u21 ,u g x= =() sec x (4) yfu= =() u ,u g x= =()xx+11b 1 l)3(log2= xya( )
5、( )+ ,33, ),( + 2 lxy 3arcsin= ( 0, 2,0 3 xy tan= l+ 2,2 kkZk )+,0 4 411+=xxy l( ) )+ ,11, )( )+,11,0 b 4. / f tf 7 (1) yx=+arcsin112; (2) 3 21log ( 1)3ayx= b 1 uy arcsin=vu1= 12+= xv 2331uy = b vualog= 12= xv5. p/ f 1 l (1) xyasinlog= 1a(2) yx= cos (3) yx=432x (4) yxx=+241b 1l() )12(,2 +kkZk ( 0, 2
6、l+ 22,22 kkZk 1,0 3l 1,4 25,0 4l()()+ ,00, +,2233b 6. / f f g $ (1) fx()=2log ( )ax g x()= 2logax (2) fx()=22sec tanx x , g x()= 1 (3) fx()= sin cos22x x+ , g x()= 1b 1f f g 5 2f f g 3f f gb 7. (1) ! , p fx x x x()+= +32 3 5321 fx()(2) !13131 += xxxxf , p b fx() 1 7 tx =+3 5 3=tx T 1)3(5)3(3)3(2)(23
7、+= ttttf 977721223+= ttt 9777212)(23+= xxxxf 2 7 txx=151=ttx T ()113113+=tttttf1412+=tt 1412)(+=xxxf b 8. ! fx()=+11 x, p , f Vr Tb ffnull ffnullnullf ffnullnullf null f 121)(+=xxxff null 322)(+=xxxfff nullnull 5332)(+=xxxffff nullnullnull b l f VV UB f B f b (, +) A 2)()( xfxf + f 2)()( xfxf f 7 2)
8、()()(xfxfxf+=2)()( xfxf + b |L ABCD V Uf 1“ yfx= ()s V U A = (,)03 B =(, )11C = (,)32 D= (,)40b 6 ( ( +=4,3823,125231,034xxxxxxy b y (, )11 O x 2 x m 1.2.8 m 1.2.9 ! V Um 1.2.8|s f fx() yfx= (), x , 02Vr Tb ( 2210,12121 1,22xxyxx x=+b Bj a a A81sY 13.610.8X (m 1.2.9) A8 5 A8 4 /A8 2 k p +1 ib Amax12
9、sinmax =B y Bx 2,0 P sin=x B2sin x i Sy P Sy sup OTy +=AB y B“ iS Sx P ABx A“ SBA = “ Bb V“/ Bb 6. d b “ SA sup S inf Sb sup S = inf S H “ S I+? i Sx SxS supinf bSS infsup sup S = inf S H “ S B L “b 107. d b/ “A/ b I 2.1.1 b 8. ! S 3|2pq5 32r2234r 32mn | sl P0r 3422+=2222rrmnmnrmn3422+rrmn rmn 9 Smn
10、S =sup b S b V S / b 115 K 1. l / kl +112nn () (.)109nn+nn51 +3321nnnull nn32 !3nn nnn! +nnnnn21)1(21111null b 1 )20( lglg0.99( 1) (0.99) (0.99)nn21 5n2H 15nn+n22 2n333(12)2nnnC= |=1,11maxN H Nn n55521321!53!3 lg112lg2Nm |=1N H Nn |=21N H Nn 13 |=81N H Nn na221)1(23nnnnaCan +=+ 3nnnnnan3)1()13(2 |=2
11、9N H Nn n i0 N P n N H xnn ni k , Pa a!)b 0 x x 1 5nxn= nx Hq kl b 2 = nnnnxn 15 nx Hq kl b 4. ! k B limnxn= a sA1Hq b limnxnk+= a ! 5limnxn= a 0 N Nn N Nn 1N 1Nn NNn 2 NNn Mxn kkk 121)2(642)12(5310+na null,2,1=n 1lim1=+laannn5 b 0lim =nna | lr =+laannn V NnN , 11+raann1110+na null,2,1=n aaannn=+1li
12、m 5 aannn=lim b nnnnnaaaaaaaa123121= null # aaannn=+1lim V aannn=lim b 12. ! (aa )i limnan12+null(1) limn1212naa nan()+null = 0 (2) limn(! )naa ann121null = 0 ( , i = 1,2,l,n)b ai 018 1 ! nnSaaa =+ null21limnaSn= 5 V ka nS Skknkn=111klimn0111limlim1111=aaSnnnSkannkknnnnkkb 2nnaaan121)!(0 nulla(3) (4
13、) nn tanarc+ nnn 212111“ b 1 | H 0G 3 GN = Nn Gnnn+31212b 2 | H 0G GaN = Nn Gnnaa=log1log b 3 |0G 2+= GN H Nn Gnn arctan b 4 | H 0G 22GN = NnGnnnnn+ 2212111“ b 2. (1) ! limnan= +( )l limnaa ann12+ + +“= +( ); (2) ! a #0 = 0 1 nlimnanlimn(aa ann121“ ) = 0b 1 ! 5+=nnalim GaNnNGn3:,0,011 b% 1N:,21NnNN
14、2121 GnaaaN+223121“b V H limnan= limnaa ann12+ + +“= b 20 2nnaaa121)ln( “naaanlnlnln21+=“ =nnalnlim V =nnnaaa121)ln(lim “ V7 limn(aa ann121“ )n n= 0b 3. (1) ! kv a ax y 05 kv xn ny(2) ! kv limxnnyn=b05 xnynnnyx kvb 1y kv xn0G N Nn Gxn b Nn Gyxnn 9 kv b xnyn 2 0 V limnyn=b N Nn bybn22 by kv xn0G “N “
15、Nn GbbGxn2,2max b| “,max NNN = Nn Gyxnn Gyxnn xnynnnyx kv b 4. (1) Stolz limn135 21 43222 23+=“ ()nn (2) pKlimn+34)12(53132222nnn“b 1limn=+32222)12(531nn“limn34)1()12(332=+nnnb 21 2limn+34)12(53132222nnn“=nlim2322234)12(313nnn + “=nlim22332)1(33)1(44)12(3+nnnnn=nlim 436124=nnb 5. Stolz (1) limnlogan
16、n= 0 ( ) a 1(2) limnnakn= 0 ( a 1 k )b 1limnlogann= limn01log =nnab 2limnnakn= limn=1)1(nnkkaannlimn)1()(11aanPnk 1n Q T V Q )(1nPk1k klimnnakn= limn=)1()(11aanPnklimn=222)1()(aanPnk=nlim“ 0)1()(0= kknaanPb 6. (1) Stolz limnxxyynnnn11= ? limnxynn= ? (2) Stolz limnxxyynnnn11i ? limnxynni ? 1 ?b I n 0
17、 , xnnn=()1 ynn= limnxxyynnnn11=nlim =1)12()1( nn limnxynnnn)1(lim =Kib 2 ?b I n 0 , xnnn=+1234 11“ () ynn=2limnxxyynnnn112212)1(lim1=nnnnKi limnxynn0= b 7. ! 0! ! 1 limnan= alimn(aa )a ann nn+ 1220“ =a1 b : 51=knnnnnnnnkaakakaaa01101+=+“ Stolz limn( )aa a ann nn+ 1220“nnnnnnkaakak011lim+=“)1(lim1=k
18、kaknnnn=1ab 8. ! , HKb 9 O(n b Aankkn=1n pnp +nlimnpa pa papnnn11 2 20+ + +=“b ! TAAnn=lim1=kkkAAa =+nnnpapapap “2211nnnnnpppAppAppAA)()()(11232121 +“ T pK pBs TK H Stolz limnnnnpapapap + “2211nnnnnnnpppAppAppAA)()()(limlim11232121 +=“= A limn11)(nnnnnppppA0= AA b 235 l 51 limnenn=+11 p/ K limnnn11
19、; limnnn+111 ; limnnn+211 ; limnnn+211 ; (5) limnnnn+2111 b 1limnnn11=nlim =+ 1)1(111111nnne1b 2limnnn+111=nlim =+ 11111111nnneb 3limnnn+211=nlim =+212211nneb 4limnnn+211=nlim =+nnn12211 1b 5 H 2nnnnnnnn+xx V n 9F yN l b ! T=01+ nnxxnx axnn=limxn+12 + xnpKZ aa += 2 NZ yN 2=a2lim =nnx b 2 n5 =10 x=nn
20、xx V 9F yN l b ! T=nx axnn=limxn+1 nx2 pKZ aa 2= NZ 6B yN 2=a0=a2lim =nnx b 3 n5 121=x ! 1kx 5 = 1+kx 121+kx 25B ,E V bn 1nx =+ nnxx1=+nnxx2102)1(2+=nnxx V 9F yN l b ! T=nx axnn=limxn+143+ xnpKZ aa 34+= NZ yN 4=a4lim =nnx b 5 n5 !101=nnnnnxxxxx V 9F yN l b ! T= (2 ) pKZnx axnn=limxn+1xn nx )2( aaa =
21、 NZ 6B yN 1=a0=a261lim =nnx b 3. w T (1) limn2335471210 +=“nn (2) limnann!= 0 (a#1) (3) nlim 0!=nnnb 1 !121745332+=nnxn“ 5 0nx 13221nx an 111nx 1111+=+nnnnxx h / yN l b !nxaxnn=lim T111+=nnnxnx pK yN eaa = 0=alimnann!= 0b 4. ! =xn+1+nnxx221, s = 1 n =123,“ x121=x f p27limnxnb O H11=x n 0nx 2n 2nx b
22、0121+=+nnnnxxxx V nx h / l b ! T =axnn=lim xn+1+nnxx221pK )2(21aaa += 2=a 2=a yN limn2=nx b 21=x n 2nx b 0121+=+nnnnxxxx V 9F l b !nx bxnn=lim T =xn+1+nnxx221pK )2(21bbb += 2=b 2=b yN limn2=nx b 5. ! = a , = b,x1x2xxxnnn+=+212 n =123,“ p b limnxn n5 w T )(2111 +=nnnnxxxx Y Tnnxx +1)(2111abxxnnn=+b )
23、()()(123121 +=nnnxxxxxxxx “=+=1021)(nkkaba limnxn32ba+= b 6. 0! ! b 7 = a , = bb a x1 1y(1) = xn+1xynn =yn+1xynn+2 n =123,“ b b l O = b Kxnynlimnxnlimnyn28 + ( a b(2) = xn+1xynn+2, = yn+12xyxynnnn+ n =123,“ b, l O = b K ( b xnynlimnxnlimnyna b 1 n5 bnnnyx 0)(1=+ nnnnnxyxxx nnyy +10)(21=nnyx byyxxann
24、nn+nx 12 nx H 1201=x n 1212+nx 1202+nnnxxx 29V h / 12 nx nx29F V7l b ! = Tlimnnx2a limnbxn=12=+12nx1212252+nnxx =+22nxnnxx22252+pKZaaa252+= bbb252+= N Z12 =a 12 =b 6 12 =a 12 =b yN 12lim =nnx b 8. ! b B = sA1Hq i b0 xnlimnxnaxnxnklimkxnk= ab A1A C sb ! b9F xnlimkxnk= a5 0 K Kk 01+ KM P MKnnn N Nnm 0nmxx b | 11=N111Nnm 011nmxx | 12mN =222Nnm 022nmxx ,“ | 1=kkmNkkkNnm 0kknmxx .“ b 0 b b bn5 b l 0 b b9 l 0 bb xnknxkmxknxknxkmx“kmx30
