Ph ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 1Bi1. Gi i h n c a dy sPh ng php gi i bi t p:BI T P M U:Bi 1. Cho dy (un) thomnnu n >v i m i n. Ch ng minh r ng l i mnnu+= +Gi i:l i mv vay l n hn mot sodng bat k ketmot sohangnao otri . mat khac unen l n hn mot sodng bat k ketmot sohang nao o.Vayl i mn nnnn nn uu+= +>= +Bi 2. Cho dy s(un)c2 1nnun+= . Tm l i mnnu+.Gi i:2 1 1Ta bi en oi :2 .1Vayl i m 2vl i m 0nn nn nnun nu un+ ++= = += =Bi 3. Bi t dy s(un) tho mn21nnun+sv i m i n. Ch ng minh r ng l i m 0nnu+=Gi i t2 21 1. Ta col i m l i m 0. Do o,cothenhohn mot sodngtuy yketmot sohang nao otri . (1)Mat khac, theo gi athi et ta co (2)T(1) va(2) suy ran n nn n nn nv v vn nu v v+ += = =s s cothenhohn mot sodng tuy yketmot sohangnao otri , ngha l al i m 0nnuu =CH NG 4. GI I H ND ng 1: Tm gi i h n c a m t dy:Ph ng php 1: Dng nh ngha tm gi i h n c a m t dy l i m 0nnu+= khi v ch khi |un| c thnhh n m t sd ng b tu ,ktsh ng no tr i.( )l i m l i m 0n nn nv a v a+ += = l i mnnu+= + khi v ch khi un c thl n h n m t sd ng l n tu, ktm t sh ng no tr i. l i m l i m( )n nn nu u+ += = +www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 2BI T P P D NG:Bi 1. Bi t dy s(un) tho mn2nu n >v i m i n. Ch ng minh r ng l i mnnu+= +Gi i:2 22Vl i mnen ncothel n hn mot sodng tuy y, ketsohang nao otriMat khac, theo gi athi et vimoi n, nen cung cothel n hn mot sodng tuy,kn nnu n uy= +>etsohang nao otri . Vay l i mnu = +Bi 2. Cho bi t l i mnnu+= vn nv u = += Bi 3. Cho dy s(un) h i t , dy (vn) khng h i t . C k t lu n g vsh i tc a dy( )n nu v .H ng d n: K t lu n dy ( )n nu v khng h i tTh t v y:( ) ( )Xet day,gi asnohoitungha l al i mval i m .Khiol i m l i mVayl i m l i mVl i m l i mVay ( ) l ahoitu, i eu nay khon n n n nn nn nn nn nn nn nn nnu v u v a u bu v av a uu b v a bv+ ++ ++ ++ + = =+ == = = ( )ng ung.Vay day khong hoitu.n nu v Bi 4. Cho dy (un) xc nh b i:3 21nnun+=+a) Tm sn sao cho131000nu 999 th cc sh ng c a dy (un) u n m trongkho ng (2,999;3,001).H ng d n:1 1) 3 9991 10001 1 1) Khi 999 3 3 3 2, 999 3, 0011000 1000 1000nn n na u nnb n u u u = < >+> < < < + < l i m 0 l i m ; l i m l i m 0n nn n n nn nA Av vv v = = = =2. nh l vgi i h n h u h n:( )( )*Gi asl i mval i m . Khio:1. l i m2. l i m . .3. l i m , 04. l i m (vi 0vimoin N )n nn nn nnn nnnnnn nnu a v bu v a bu v a bu abv bu a u+ +++++= = = == == > e3. nh l vgi i h n *1. Neul i m val i m thl i m 02. Neul i m 0, l i m 0va 0,thl i m3. Neul i m val i m 0 thl i mnn nn n nnnn n nn n nnn n n nn n nuu a vvuu a v v nvu v a u v+ + ++ + ++ + += = == > = > eN = += + = > = + N u bi u th c c d ng phn th c tsv m u sch a luth a c a nth chia tv m u cho nk v i k l mcao nh t. N u bi u th c ch a cn th c ( d ng3 3; A B A B ) c n nhnm t l ng lin hi p a vd ng cb n.www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 53 233 22 35 133 5 1 3l i m l i m6 4 5 2 2 6 4 52n nn nn nn n nn n n+ + + += =+ + ++ + +Bi 2. Tnh222 1 5l i m1 3nn nn+++.Gi i:22221 1 522 1 5 0l i m l i m 01 3 1 33n nn nn n nnn+ ++ +++= = = Bi 3. Tnh( )2 2l i m 7 5nn n++ +Gi i( )2 22 22 2 2 27 5 2l i m 7 5 l i m l i m 07 5 7 5n n nn nn nn n n n+ + ++ + + = = =+ + + + + +Bi 4.Tnh( )2 2l i m 3nn n n++ Gi i:( )2 22 23 3 3l i m 3 l i m l i m23 31 1n n nnn n nn n nn+ + ++ = = =+ ++ +BI T P P D NG:Bi 1. Tnh cc gi i h n sau:( )( )( )2 222 310 1 110 1 14 22Tong4 1 1 2) l i m ) l i m ) l i m1 3 2 2 5... Tnh gi ihan:l i m...Tnh gi ihan sau:2 1d)l i m q)uat:2 1 3 2n n nm mm mp pnp pnn n n na b c nn n na n a n a n ab n b n b n bn nen n n+ + +++| | |+ + +\ .+ + + ++ + + + ++ +( ) ( )3 252 3 1l i m1 4nn nn+ +p s :27) 2 ) 0 ) ) 1 )4a b c d e + Bi 1.1( )2Tnh:l i m 1nn n n+ + +Gi i:( )221 1Tnh:l i m 1 l i m( ) 1n nn n n nn n+ +| | + + = = | |\ .www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 6Bi 2. Tnh cc gi i h n:4 2 2 2 22 23 32 7 3 1 1 3 14) l i m ) l i m ) l i m2 3 1 22) l i m2n n nnn n n n n na b cn n n nn ndn+ + +++ + + + + ++p s :32) ) 3 1 ) 0 ) 22a b c d Bi 3. Tnh gi i h n sau:( )( )1 1 1113 2 3 2 4.3 7) l i m ) l i m ) l i m3 2 1 2 2.5 72 35 1) l i m ) l i m5 12 3n n n n nn n n n nn n nnnnn nn n na b cd e+ + ++ + +++ + + ++ + + ++ +p s :1) 3 ) ) 7 ) )13a b c d e + Bi 4. Tnh cc gi i h n sau:( ) ( ) ( )( ) ( )( )3 2 3 222 2 223 32 2) l i m 1 ) l i m 3 2 ) l i m 24 1 2 1) l i m ) l i m ) l i m 1 221) l i m 2 ) l i m2 4n n nn n nn na n n b n n n c n n nn nd n n n e f n n nn n ng n n n hn n+ + ++ + ++ ++ + + + ++ + + + ++ +p s :7 2 1 3) 0 ) ) ) )1 ) )3 )2 3 2 2a b c d e f g h Bi 5.Tnh cc gi i h n sau:2 21 2 3 ... 1 2 3 ...) l i m ) l i m1n nn n na bn n n+ ++ ++ + + ++ ++ +( )2221 1 1 1 1 ...) l i m ... ) l i m vi 1, 11.2 2.3 3.4 ( 1) 1 ...1 3 ... 2 1) l i m2 11 1 1 1) l i m ...1.2.3 2.3.4 3.4.5 ( 1) 2nnn nnna a ac d a bnn b b bn nen nfnn n+ +++| | + + + ++ + + + < < |+ + + + +\ .++ + + +| | | + + + + |+ +\ .www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 7( )( )( )2 2 24*2 32 2 2) l i m 1 1 ... 12.3 3.4 1 21 1 1 1) l i m ...1.3 3.5 5.7 (2 1)(2 1)2.1 3.2 ... 1) l i m1 1 1) l i m ...2 1 2 3 2 2 3 ( 1) 11 3 5) l i m ...2 2 2nnnnngn nhn nn ninkn n n nl+++++| || || | | | | |+ +\ .\ .\ .| |+ + + + | +\ .+ + + +| |+ + + | |+ + + + +\ .+ + +2 12nn | | + |\ .H ng d n v p s :( )22 22121 2 3 ... 1) l i m l i m l i m1 11 2 2n n nnn nn n n n nan n n nn n+ + +| | + |+ + + + +\ .= = =+ + + ++ +b)121 1 1 1 1 1 1 1 1 1 1) Ta co:1 ; ; ; ...;1.2 2 2.3 2 3 3.4 3 4 ( 1) 11 1 1 1 1Suy ra:l i m ... l i m 1 11.2 2.3 3.4 ( 1) 1n ncnn n nnn n+ += = = = + +| | | |+ + + + = = ||+ +\ . \ .d)111l i m1 11nbaSab+= =e)( )2 21 2 11 3 ... 2 1 12l i m l i m2 2 1 2 1n nn nnn nSn n n n+ ++ ++ + = = =+ + + +f)( )( ) ( ) ( )( )( )( ) ( )( )( ) ( )( )1 1 1 1Sdung:2 1 2 1 1 21 1 1 1 1 1Vay:...1.2.3 2.3.4 2 2 . 1 2 1 21 1 1 1 1 1 1 1Vay l i m ... l i m1.2.3 2.3.4 3.4.5 2 2 4 ( 1) 2 1 2n nk k k k k k kn n n n nnn n n n+ + (= (+ + + + + ( (+ + + = (+ + + + ( | |( | + + + + = =( |+ + + + (\ . g)( )( )( )( )1 22Ta thay: 11 1k kk k k k + = +www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 8( ) ( )( )( )( )( )( )( )( )( )2 2 2 2Vay: 1 1 ... 1 ... 12.3 3.4 . 1 . 11 2 1 21.4 2.5 1 3. ... ...2.3 3.4 3 1 1 12 2 2 1Vay l i m 1 1 ... 12.3 3.4 3 1 2nk k n nk k n nnn k k n nn n+| | | || || | || | | ||+ +\ .\ .\ . \ . + +| | += = |+ + +\ .| || || | | = | | |+ +\ .\ .\ .1 1 1 1 1 1 1 1 1 1) ... 1 ...1.3 3.5 5.7 (2 1)(2 1) 2 3 3 5 2 1 2 11 1 11nenl i m2 2 1 2nnnhSn n n nSn+ (= + + + + = + + + ( + + (= = (+ ( ) ( ) ( ) ( )( ) ( )( )( ) ( )( )2 2 2 2 2 223 3 3 2 2 2224 4 4) Ta co:2.1 3.2 ... 1 1 1 1 2 1 2 ... 11 1 2 11 2 ... 1 2 ....2 61 1 2 11l i m l i m4 4 6nnnn ni S n n n nn n n n nS n nn n n n nSn n n+ += + + + + = + + + + + + (+ + += + + + + + + + = +( ( (+ + + (= + = ( ( )( )( ) ( )( )221 11 1 1) Ta co:1 1 11 11 1 1...2 1 2 3 2 2 3 1 11 1 1 1 1 11 ... 1 l i m 12 2 3 1 1nnnn n n nkn n n n n nn n n nSn n n nSn n n++ += = + + + ++ += + + ++ + + + += + + + = =+ +2 32 2 3 3 112 1 1 1 2 11 3 5 2 1) Ta co:...2 2 2 21 1 3 1 5 3 2 1 2 3 2 1...2 2 2 2 2 2 2 2 21 11 1 1 1 2 1 1 2 1 1 1 2 12 2... 11 2 2 2 2 2 2 2 2 2 2121 1 1Suy ra:12 2 2n nn n n n nnn n n n nn nnl Sn n nS Sn n nS+ + + += + + + +| | | | | | = + + + + |||\ . \ . \ . = + + + + = + = + = + ( )2 1 32 1 1 2 132 2 2 22 2Mat khac:. Mal i m 0 l i m 01 1 2 21 1Vayl i m 3n n n n nn n nn nnnn nSn n nn nS + + ++ = += < = = +=www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 9BI T P M U:Tnh2 2 21 2l i m ....1 2nnn n n n+| |+ + + |+ + +\ ..Gi i:Ta th y:( )( )( )( )( )( )2 2 2 22 2 2 2 2 222 2 2222 2 21 2 1 2 ... 1....2 1 211 2 1 2.... ...1 2 1 1 12 111 1 2....2 1 22 111l i m22 11 2 1l i m ....2 1 2nnn nn n n n n nn nn nVan n n n n n nnn nnVayn n n nnn nMannVayn n n n+++ + ++ + + > =+ + + +++ + + s + + + =+ + + + + +++s + + + s+ + +++=+| |+ + + = |+ + +\ .BI T P P D NG:Bi 1. Tnh gi i h n c a cc gi i h n sau:( )222 2 21 1 3si n 4 osn si n) l i m ) l i m ) l i m2 3 n+1 3n+41 3nsi n2 os2n) l i m ) l i m3n+1 cosn+5n1 1 1) l i m ...1 2nn n nnn nnn c n na b cnn cd efn n n n+ + ++ ++| | + + |\ . ++| |+ + + | |+ + +\ .p s :1 3) 0 ) 0 ) )0 ) )13 5a b c d e fBi 2. Cho 2 dy s(un) v (vn). Ch ng minh r ng n u l i m 0 van nv u v = sv i m i nth l i m 0nu =Ph ng php 3. Dng nguyn l k p.Cho ba dy s(un), (vn) v (wn). N uvimoinn n nu v w s sV l i m l i m ( ) th l i mn n nu w L L v L = = e = Rwww.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 10H ng d n:l i m 0cothenhohn mot sodng betuy y, ketmot sohang nao otri . (1)V vavimoin, nen vimoin(2)T(1) va(2) suy racung con nn n n n n nnv vu v v v u vu= s s sthenhohn mot sodng tuy y, ketmot sohangnao otri , ngha l al i m 0nu =p d ng: Tnh gi i h n c a cc dy sc sh ng t ng qut nhsau:21 ( 1) 2 ( 1)) ) )! 2 1 2 11n nn n nna u b u c un n n = = = +) (0, 99) cos ) 5 cosn nn nd u n e u n = = p s :) 0 ) 0 ) 0 ) 0 ) a b c d e + D NG 2: Ch ng minh m t dy sc gi i h n:Ph ng php:1. p d ng nh l Vy strax :- N u dy s(un) tng v b ch n trn th n c gi i h n.- N u dy s(un) gi m v b ch n d i th n c gi i h n.2. Ch ng minh m t dy stng v b ch n trn ( dy stng v bch n d i) b i sM ta th c hi n: Tnh m t vi sh ng u tinc a dy v quan st m i lin h d on chi u tng (chi u gi m)v sM.3. Tnh gi i h n c a dy sta th c hi n theo m t trong hai ph ngphp sau:* Ph ng php 1: t l i mnnu a+= T1l i m l i m ( )n nn nu f u++ +=ta c m t ph ng trnh theo n a. Gi i ph ng trnh tm nghi m a v gi i h n c a dy (un) lm t trong cc nghi m c a ph ng rnh. N u ph ng trnh cnghi m duy nh t th chnh l gi i h n c u dy c n tm. cnn u ph ng trnh c nhi u h n m t nghi m th d a vo tnhch t c a dy s lo i nghi m. Ch : Gi i h n c a dy sn u c l duy nh t.- Ph ng php 2: Tm cng th c t ng qut un c a dy sb ng cch d on./ Ch ng minh cng th c t ng qut un b ng ph ng php quyn p ton h c. Tnh gi i h n c a dy thng qua cng th c t ng qut .www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 11BI T P M U:Bi 1. Ch ng minh dy (un)b i cng th c truy h i1122 vin 1n nuu u+== + >.Ch ng minh dy c gi i h n, tm gi i h n .Gi i:Ta c:1 12 va 2 , 0 vi n Nn n nu u u u+= = + > e Ta ch ng minh : 2 vi (1)nu n N < e1Vin=1, ta co 2 2 th (1) ungGi asbat bat ang thc ung vin=k th2.2,knuuVay u n N= < < t l i m th 0 a 2nnu a+= s sTa c:1 122 l i m l i m 22 2 0 1hoac=2V0nenl i m 0. Vay l i m=2Trong l igi aitren, ta aap dung tnh chat sau:"NeuLu l y:n n n nn nn n nn nu u u ua a a a a au u a u+ ++ ++ += + = += + = = > = >1i mthl i m "n nn nu a u a++ += =Bi 2. Cho dy (un)b i cng th c truy h i11212nnuuu+= = . Ch ng minh r ng dy s(un) c gi i h n v tm gi i h n .Gi i:Ta c :1 2 3 411 2 3 4; ; ; . Tota doan:(1)2 3 4 5 1Chng mi nh doan tren bang quy nap:1 1Vin=1, ta co:(ung)1 1 2Gi asang thc (1) ung vin=k (k 1), ngha l a1nknu u u u unukuk= = = = =+ = =+ > =+.www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 121*1 1 1Khiota co , ngha l aang thc (1)2 221cung ung vin=k+1.Vay, .1Tota col i m u l i m 11kknnkuk u kknu nnnn++= = = ++ = e+= =+NBI T P P D NG:Bi 1. Ch ng minh dy (un) v in dau can2 2 ... 2 2nu = + + + +_ l dy h i t .Ph ng php: Xt dy (un) tng (ho c gi m), xt (un) b ch n trn (ho c b ch n d i)Ch : tm gi i h n c a dy cho b i cng th c truy h i ta dng cc ph ng php.1. Tm cng th c t ng qut ( d a vo ph ng php c nu ph n ki n th cdy s ).Tnh gi i h n un.2. Tm ( )1l i m l i mn nn nu f u++ += . Gi i ph ng trnh tm l i mnnu a= Tm gi i h n.Bi 2. Cho dy truy h i1103( 2)4nnuuu n= += >. Tmgi i h n c a dy.H ng d n v p s :1122211103 114 415 1116 4...1141bang phng phap quy nap chng mi nh 141Vay l i m 1 14nnnnnnuuuuu+=| |= = |\ .| |= = |\ .| |= |\ .| |= |\ . (| | ( = | (\ . www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 13Bi 3. Cho dy truy h i1121( 2)2= += >nnuuu n. Ch ng minh dy (un) c gi i h n, tm gi ih n .H ng d n v p s :Cch 1:112 1Doan2 12 1l i m l i m 12 1nn nnn nn nuu+ ++=+= =Cch 2: Ch ng minh dy gi m v b ch n d i. Gis1l i m ,tm a1l i m l i m 12l i m 1nnn nn nnnu aau u a au++ ++=+= = = ==Bi 4.a) Cho dy truy h i1121( 1)2+= += >nnuuu n. Ch ng minh dy (un)c gi i h n v tmgi i h n .b) Cho dy (un) xc nh b i:( )10 111 ( 1)4+< < > >nn nuu u n. Ch ng minh dy (un)c gi i h n v tm gi i h n .H ng d n v p s :( ) ( )*1 1 1) *Chng mi nh (u ) l aday tang vabchan trenTa co: 0 1,Ap dung bat ang thc coi :11 2 1 2 1 ,4Vay ( ) l aday tang vabchan tren th ( ) th dnnn n n n n nn nbu n Nu u u u u u n Nu u+ + +< < e+ > > = > e( ) ( ) ( )21 1ay cogi ihan*atl i m , 01 1 1 1 1Taco:1 l i m 1 1 04 4 4 2 21Vayl i m2nnn n n nnnnu aau u u u a a a au++ +++= >| | ( > > > s = | \ .=www.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 14Bi 5. Cho dy (un) xc nh b i1 11 2va 02n nnu u uu+| |= + > | |\ .a) Ch ng minh r ng 2 vimoin 2nu > >b) Ch ng minh dy (un)c gi i h n v tm gi i h n .H ng d n v p s :( )*1 111 2) Ta co:0, 0,2 dung bat ang thc Cosi :1 2 2. 2 , 1,2Suy ra u 2, 2,) Ta co: u 2, 2,nen l aday bchan diXetn n nnn n nn nnn na u u u u n NuApu u u n nu un n Nb n n N u++| |> = + > e | |\ .| |= + > = > e N | |\ .> > e> > e2*1 121 11 2 1u u 1 0, 2, nen u ,2 u 2*atl i m , 2. Ta co:2 1 2 1 2 1 2l i m l i m 22 2 22Vaynn n n n n nn nnnn n n nn nn nuu u n n N u n Nuu aaau u u u a a au u aa+ +++ ++ +| | | | = + = < > e < e || ||\ . \ .= >

| | | |= | |= + = + = + = || | || \ . = \ . \ .l i m 2nnu+=Bi 6. Ch ng minh dy (un) c cho b i cng th c*cos .nu n n = eN . Ch ng minhdy khng c gi i h n.H ng d n:( ) ( )( ) ( )( )2Gi asl i m l i m cos l i m cos 2 l i m cos 2 cos 02 l i m si n 1 si n1 0 l i m si n 1 0 l i m si n 0mat khac:si n 1 si n os1 cos si n1, Suy ra l i m cos 0Suy ra: l i m cosnn n n nn n nnnu n a n a n nn n nn nc n nn+ + + ++ + +++ (= = + = + = + = + = =+ = + =( )2si n 0,vol yVay day so( ) vi coskhong cogi ihan.n nnu u n+ ==Bi 7. Ch ng minh cc dy sau h i t :2 2 22 31 1 1) 1 ... ;2 31 1 1) 1 ... ;2 3nn na n Nnb n Nn= + + + + e= + + + + eH ng d n:a) Ta th ywww.VNMATH.comPh ng php gi i bi t p gi i h n dy s 01234332133Tr n nh C - Tr ng THPT Phong i n 152 2 22 2 21 1 1Day1 ... l aday tang, ta chcan chng mi nh day bchan.2 31 1 1 1 1 1 11 ... 1 ... 2 21.2 2.3 ( 1) 2 3Vay day hoitu.nnn n n n = + + + ++ + + + < + + + + =