B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 1Chng 2 PHP TNH VI PHN HM MT BIN 1. GII HN LIN TC I. Dy s - Gii hn dy s. 1. Dy s 1.1 nh ngha Dy s l mt tp hp cc s c vit theo mt th t xc nh: { }1 2, 3, ,..., ,...nx x x x . ch dy s , ngi ta thng dng k hiu { }1nnx=hay gn hn{ }nx . Trong chng ny, ta ch xt cc dy s thc. Dy s thc l mt nh x : ( ):

=

nfn f n x K hiu{ }nnxhay{ }nx . Lc : n c gi l ch s. nxc gi l s hng tng qut ca dy. Ch : Dy s cn c th xc nh bi cng thc tng qut 1 21 21, 22 , 3n n nx xx x x n '= =11!1 = 1+ Ghi ch: Ta thng xt dy s thc l nh x t * vo . V d 1. 11 1 1 1) 1, , ,..., ,...2 3nan n= = ` ` ) ); ( ){ }( ){ }) 1 1,1, 1,1,..., 1 ,...n nb = ; { } { }2 2) 1, 4, 9,..., ,... c n n = ; 1 2 3) , , ,..., ,...1 2 3 4 1n ndn n = ` `+ + ) ). Dy s{ }nx gi l tng nu *1, n nx x n+< , gi l gim nu *1, n nx x n+> . Trong v d 1, dy a) l dy s gim, dy c) l dy s tng. Dy s tng v dy s gim c gi l dy s n iu. Dy s{ }nxgi l b chn trn nu tn ti mt s M sao cho *,nx M n ; gi l b chn di nu tn ti mt s m sao cho *,nx m n ; gi l b chn nu n va b chn trn va b chn di. V d 2. Trong v d 1 Dy a) l dy s gim, n b chn di bi 0 v b chn trn bi 1; Dy b) khng phi l dy s n iu, n b chn di bi -1 v b chn trn bi 1; B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 2Dy c) l dy tng, n b chn di bi 1 nhng khng b chn trn, do n khng b chn; Dy d) l dy s tng, n b chn di bi 0 v b chn trn bi 1. 2. Cc dy s c bit 2.1 Dy s cng 2.1.1 nh ngha L mt dy s tho mn iu kin: hai phn t lin tip nhau sai khc nhau mt hng s. Chng hn, dy s 3, 5, 7, 9, 11, ... l mt cp s cng vi cc phn t lin tip sai khc nhau hng s 2. Hngssaikhcchungcgilcngsaicacpscng.Ccphntcan cng c gi l cc s hng. 2.1.2 S hng tng qut Nu cp s cng khi u l phn t u1 v cng sai l d, th s hng th n ca cp s cng c tnh theo cng thc: n 1u u (n 1)d = 2.1.3 Tng Tng ca n s hng u ca cp s cng c gi l tng ring th n. Ta c: [ [1 1 nn 1 2 nn 2a (n 1)d n(a a )S a a ... a2 2 = = =2.2 Dy s nhn 2.2.1 nh ngha L mt dy s tho mn iu kin t s ca hai phn t lin tip l hng s. T s ny c gi l cng bi ca cp s nhn. Cc phn t ca cp s nhn cn c gi l cc s hng.Nh vy, mt cp s nhn c dng 2 3a, ar,ar , ar ,...Trong r 0 l cng bi v a l s hng u tin 2.2.2 S hng tng qut S hng th n ca cp s nhn c tnh bng cng thc n-1na ar =trong n l s nguyn tha mn n>1 Cng bi khi l 1n 1n nn 1a ar , ra a 1 = =

( ) trong n l s nguyn tha mnn 1 2.2.3 Tng Tng cc phn t ca cp s nhn : B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 30 12 1 23 34 45 nk 0 1 2 nnk 0S ar ar ar ar ... ar== = Hay n 1na(1 r )S1 r= 2.3 Dy Fibonacci Dy Fibonacci l dy v hn cc s t nhin bt u bng hai phn t 0 v 1, cc phn t sau c thit lp theo quy tc mi phn t lun bng tng hai phn t trc n. Cng thc truy hi ca dy Fibonacci l: n0 , khi n 0F : F(n) : 1 , khi n 1F(n 1) F(n 2) , khi n 1'=1111= = =!11 11+ 3. Gii hn ca dy s Tr li dy d) ca v d 1. Biu din hnh hc ca n c cho hnh sau:

Tanhnthyrngkhincnglnth nx cnggn1,tclkhongcch1nx cng nh, n c th nh bao nhiu cng c min l n ln. Ta ni rng dy { }nx gn ti 1 ( hay c gii hn l 1) khi n dn ti v cng. Ta c nh ngha sau:nh ngha: S a gi l gii hn ca dy s{ }nx nu vi mi sdng b ty cho trc, tn ti mt s t nhin 0nsao cho vi mi 0n n >th nx a < .Ta vit:limnnx a= hay nx a khin . Khi , dy s { }nx c gi l hi t. Dy s khng hi t c gi l phn k.Ch : Ch s 0nph thuc vo, nn ta c th vit( )0 0n n = . V d 3.a) Chng minh 1lim 02nn= . Thtvy,chotrc0 > ,taschrarngtmc( )*0n cho 010 ,2n nx n n = < > . Ta c, 12n , tc l khi 21log n> . Vy ch cn chn( )0 21log n =th vi 0n n > ta c0nx < . B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 4b) Dng nh ngha chng minh rng n4n 3limn 1+ 4. Cc Tnh cht v nh l v gii hn dy s Dng nh ngha gii hn ca dy s, c th chng minh c cc nh l sau: nh l 1. a) Nu mt dy s c gii hn th gii hn l duy nht. b) Nu mt dy s c gii hn th n b chn. Ch thch: Mnh b) ca nh l 1 l iu kin cn ca dy s hi t. T suy ra rng nu mt dy s khng b chn th n khng c gii hn. Chng hn, dy c) trong v d 1 khng c gii hn v n khng b chn. nh l 2. Nu cc dy s{ }nx v { }nyu c gii hn (lim ; limn nn nx a y b ) th i)( ) lim lim limn n n nn n nx y x y a b = = ii)( ) lim . lim . lim .n n n nn n nx y x y a b = =iii) limlimlimnn nnn nnxx ay y b= = ( vi iu kinlim 0nny ). V d 4. Tnh gii hn cc dy s sau nn n 2nnnn n 2nnnn n 2nn1 1 aa) a , b limn n b1 1 bb) a , b limn n a1 1 ac) a , b limn n b= = = = = = d) ( ) n 1nn nnn1 1 aa , b limn n b= = Ch : Trong tnh ton v gii hn, c khi ta gp cc dng sau y gi l dng v nh 0, ,0. , ,...0 . Khi khng th dng cc kt qu ca nh l 2, m phi dng cc php bin i kh cc dng v nh . Chng hn, 222 1lim3 5nn nn+ ++ c dng . Ta bin i: 22221 122 1 2lim lim53 5 33n nn nn nnn + ++ += =++.4.1 Tiu chun tn ti gii hn nh l 3. Cho 3 dy s{ }{ }{ } , ,n n nx y z . Nu: a) *,n n nn x y z ; b)lim limn nn nx z a = =th dy { }ny c gii hn vlimnny a= . nh l 4. a) Nu dy s tng v b chn trn th n c gii hn. B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 5b) Nu dy s gim v b chn di th n c gii hn. nh l 5. Dy s nxc gi l dy c bn ( hay dy Cauchy) nu vi mi0 tn ti s n0 >0 sao cho n mx x n0. ngha: K t mt lc no tr i hai phn t bt k ca dy s gn nhau bao nhiu cng c. 4.2 Cc v d v gii hn ca dy sV d 5. Cho dy s { }nxvi 3 59 4nnxn=+. Chng minh 1lim3nnx= . Vi k no th xk nm ngoi khong 1 1 1 1;3 1000 3 1000L| |= + |\ . Ta c 55333 5 1lim lim lim4 4 9 4 39 9n n nnn nnnnn n | ||\ = = =+ | |+ + |\ . Khong cch t xn n 13 bng ( ) ( )1 3 5 1 19 193 9 4 3 3 9 4 3 9 4nnxn n n = = =+ + +; x nm ngoi khong L khi v ch khi 1 13 1000x >hay ( )19 13 9 4 1000 n>+. Do 18988 770327 27n < = . Vy cc s ca dy nm ngoi khong L l x1, x2, , x703. V d 6. Chng minh rng 2lim 0!nnn= . Ta c ( )33s2 2.2...2 2 2 2 2 1 1 1 4 12.1. . ... 2.1. . . ...! 1.2.3... 3 4 3 2 2 2 3 2nnnn n n| |= = < = |\

. V 31lim 02nn| |= |\ nn 2lim 0!nnn= . V d 7. Tnh cc gii hn sau: a) 223 5 4lim2nn nn+ ++b) 3223 2lim4 2 7nn nn n| | + |+ +\ Gii.a)Ta c 22225 433 5 4lim lim 3221n nn nn nnn + ++ += =++. B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 6b)Ta c 3322221 233 2 3 27lim lim2 74 2 7 4 644n nn nn nn nn n | |+ || | + | |= = = |||+ +\ \ |+ +\ . V d 8. Tm gii hn ca cc dy s{ }nx sau:a)2 3 1nx n n = + b) 3 2 3nx n n n = +c) 23 41nn nxn n n+ +=+ . Gii. a)Khin ,2 3 1nx n n = + cdngvnh.Munkhdngv nh y, ta nhn t v mu ca xn vi lng lin hp2 3 1 n n + + , ta c: ( )( )( ) ( )2 22 3 1 2 3 12 3 1lim lim lim2 3 1 2 3 1414lim lim2 3 1 2 3 1 1nn n nn nn n n nn nxn n n nnnn nn n n n + + + + = =+ + + + ++= = = ++ + + + b)Tac 2 3 311 n n nn| | = |\ khin ,vvy 3 2 3nx n n n = + cdng . Nhn t v mu ca xn vi lng lin hp ( )23 2 3 2 3 23n n n n n n + , ta c: ( )( )( )( )23 3 2 3 2 3 2 3 2323 2 3 2 3 2322 23 2 3 2 3 233 3lim lim1 1lim lim31 11 1 1nn nn nn n n n n n n n nxn n n n n nnn n n n n nn n | | + + |\ = += = =| | + + |\ c)Ta c 222433 44 4 4 4 42 21 11 1111.1 1 1 11 1nnn nn nn nx nn n nnn n n n| |+ + |+ ++ +\ = = =| |+ + + |\ . Do 244 421 11lim lim .1 11nn nn nx nn n + += = ++ . V d 9. Tm gii hn ca cc dy s{ }nx sau:a) nsin nlimn B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 7b) 2n1 4lim 2 3n n 1 1 ( )( ) c) ( )( )23 2n2n 1 n 3n 2lim4n n 1 d) ( )nlim n n 1 n4.3 Gii hn m rng nnnnnnlimxlimxlimx=== V d 10. ( )( )( )2n2n2nn2na) limnb) lim n 5c) lim n 5nd) lim 1 n Gii. a) Ta c 2nlimn= 4.4 Mt s gii hn c bit( )nnnnnnn1lim 1 en1lim 0( 0)nlim n 1lim a 1 a 0 1 =

( )= == nn0 , 0 q 1limq , q 11 , q 1'< chotrc,utntimts 0 > saochokhix a < th( ) f x A < ,khiul( ) limx af x A= hay( ) f x A khix a . V d 13. Chng minh rng( )1lim 2 1 3xx+ = . Ta cn ch ra rng nu cho trc s0 > , th tm c s0 >sao cho2 1 3 x + . Tm gii hn ca( ) f x ? Ta thy( )0lim 0xf x=v( )0lim 1xf x+= .Do ( ) f xkhng c gii hn khi0 x . V d 15. Tnh gii hn cc hm s sau khi0 x : B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 9a) xf (x)x=b) 1f (x)x=V d 16. Tnh gii hn 1 pha, 2 pha cc hm s sau: 2xxx1a) limx4x 1b) lim2x 5xc) limx 1 ( )xxxx 02x 32 3d) lim2 31e)limx1f ) limx 3 1x 1x 11x 1x 1g) lim2h) lim2x x 2l) f (x)3x x 2' 11=!11+ Nhn xt: Hm s c th c gii hn mt pha nhng khng phi lc no cng c gii hn 2 pha suy ra gii hn khng phi tn ti i vi mi hm s 2.Cc php ton v gii hn nh l 5. Gi s( ) limx af x A= ,( ) limx ag x B= . Khi : i)( ) ( ) ( )limx af x g x A B = ii)( ) ( ) ( )lim . .x af x g x AB=iii) ( )( )limx af x Ag x B= , nu0 B . iv) nnnx a x alim f(x) = l imf(x) = A; A 0 , n chn v) kk kx a x alimf (x) limf (x) A , k l= = l l. vi) x alim f (x)f (x) Ax alimb b b , b 0= =. vii)[ [( ) b b bx a x alim log f (x) log limf (x) log A(A 0, 0 b 1or b>1) = =< < . Ch : Trong qu trnh tm gii hn ca hm s ta nu gp mt s cc dng v nh sau: 0 00 0; 0. ; ; ; ; ;1 ; 0 , ,...1 100 . th phi tm cch bin i kh chng. V d 17. a) ( )2 22 2 2 2 222 2 2 2limsin limsinsin 1 4lim3 1 lim3 lim lim1 3 2 4 lim 3 13 12 2x xxx x x xx xxx x x x x x = = == =+ + + + + | |+ |\ b) ( )( )( )22 222 222 2lim( 3).lim 331.1 1lim5 2 lim 5 lim2 10 2 8x xxx xx xxx x = = = B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 10 c) ( )( )( )33333lim 33lim 02 lim 2xxxxxx x= = V d 18. a) Xt 211lim .1xxx y ta gp dng v nh 00. Khi1, x c th xem1, x Ta khai trin ( )( )21 1 111 1x x xxx x + = = + . Do ( )21 11lim lim 1 21x xxxx = + =. b) Tnh 328lim2xxx. V( )( )3 28 2 2 4 x x x x = + +nn ( )322 28lim lim 2 4 122x xxx xx = + + = V d 19. Tnh cc gii hn sau: ( )( )5 3x +4 3x +23xa) lim 7x 4x 2x 9b) lim x 4x 2x 94x xc) lim2x 5 1

( ) ( )3x +2x6 3x +23x + x33x 5d) lim6x 8x 2e) lim3x 6f ) lim x 5 x2x 5 x 0g) f (x) , lim f (x), lim f (x)3 5xx 01 4x x '1 kx k thtani( ) x lVCBcpksovi VCB( ) x B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 14 Nu( )( )limx axx khng tn ti, ta ni rng khng th so snh hai VCB( ) x v( ) x . V d 26. a) 1 cos x v 2x u l nhng VCB khi0 x . V 20 0 0 0sin sin1 cos 12 2lim lim limsin .lim . 02 2 22x x x xx xx xxx x = = =Nn 1 cos x l VCB bc cao hn 2x. b) 1.sin xx v 2x l nhng VCB khi0, x V 0 0 01 1sin sin1 1lim lim limsin2 2 2x x xxx xx x = =nhng khng tn ti 01limsinxx nn 1sin xx v 2x l hai VCB khi0 x khng so snh c vi nhau. c) 1 cosx v x2 l hai VCB ngang cp khi0, x v do 1 cosx cng l VCB cp hai so vi x2 , v 22 2x 0 x 0x2sin1 cosx 12lim limx x 2 = =d) sinx v x2 u l nhng VCB khi0, xv 2 2x 0 x 0 x 0sinx x 1lim lim limx x x = = =nn sinx l VCB cp thp hn x2 hay x2 l VCB cp cao hn sinx. b)V cng b tng ng nh ngha: Hai VCB khix a gi l tng ng vi nhau nu ( )( )lim 1x axx=, K hiu :( ) ( ) x x . Nu( ) 0 x khix a th : ( ) ( )( ) ( )sin ,sin ,x xacr x x ( ) ( )( ) ( ), tg x xarctg x x 2(x)1 cos (x) ~2(x)ln(1 (x)) ~2'1 1 111!11 111+ [ [k( x)1 (x) 1 ~ k (x)e 1 ~ (x)'1 1!1 1+ nhl8:Nu( ) x v( ) x lhaiVCBkhi( ) ( ) ( ) ( )1 1, , x a x x x x khi x a th ( )( )( )( )11lim lim .x a x ax xx x =Tht vy, v( ) ( ) ( ) ( )1 1, , x x x x ta c B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 15 ( )( )( )( )1 1lim 1, lim 1x a x ax xx x = =Do : ( )( )( )( )( )( )( )( )1 11 1lim lim . .x a x ax x x xx x x x | |= | |\

( )( )( )( )( )( )( )( )1 1 11 1 1lim .lim .lim lim .x a x a x a x ax x x xx x x x = =nhl9:(quytcngtbccVCBbccao).Nu( ) ( ) , x x lccVCBkhi ( ) , x a x l VCB bc cao hn( ) x th khix a ( ) ( ) ( ). x x x + Tht vy, ta c ( ) ( )( )( )( )( )( )lim lim 1 1 lim 1x a x a x ax x x xx x x | | += + = + = | |\ V d 27. Chng minh rng 2 3sin x x x x + khi0 x . Khi0 x th 3 34 4sin sin ; x x x x = 3 3 32 3 22 2 4x x x x x x + = + = . V bc ca 2xcao hn bc ca 32x . Do 2 3sin x x x x + khi0 x . V d 28. Tnh cc gii hn sau: a) 2 20sin2 arcsinlim ;3xx x arctg xx+ b) 3 2 43 2 301 cos 2sin sin 3lim6sin 5xx x x x xtg x x x x + + +

Gii.a) Ta c 2 2 2 2sin2 arcsin 2 2 x x arctg x x x x x + + = khi0 x Do 2 20 0sin2 arcsin 2 2lim lim3 3 3x xx x arctg x xx x + = =b)Ta bin i t s: 3 2 4 2 3 2 41 cos 2sin sin 3 2sin 2sin sin 32xx x x x x x x x x + + = + + 23 42 2 3 22xx x x x| |+ + |\ khi0 x Cn mu s tng ng vi 3 2 36 5 x x x x x + khi0 x Vy ta c: 02lim 2xxx= . V d 29. Tnh cc gii hn sau s dng cc VCB tng ng B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 16 22 32 3x 12xx 022x 0x 010 8 2 23 x 7 8x 03 4 x 1x 1ln(1 x 3x 2x )a) limln(1 3x 4x x )e 1b) limln(1 4x)sin 3xc) limln (1 2x)1 2x 1d) limtg3xx 7x x ln(1 2x )(1 cos4x)e) limsin x.(e 1).arctg(3x)+x x(x 1) sin (x 1).(e 1)f ) lim1 ( ( )( ) ( )36 92x 1) 1 arcsin x-1 x 1 l l l c)So snh cc VCL Gi s( ) F xv( ) G xl hai VCL khix a . Nu ( )( )limx aF xG x= , ta ni ( ) F xl VCL bc cao hn( ) G xkhix a Nu ( )( )lim 0x aF xG x= , ta ni ( ) F xl VCL bc thp hn( ) G xkhix a Nu ( )( )( ) lim 0,x aF xAG x= , ta ni( ) F xv( ) G xlnhng VCL cng bc. Nu ( )( )lim 1x aF xG x= ,tani( ) F x v( ) G x lhaiVCLtngngkhix a khu ( ) ( ) F x G x khix a . Cng nh i vi cc VCB, ta d dng chng minh ccc nh l sau. nh l 10: Nu( ) F xv( ) G xl hai VCL khix a ,( ) ( ) ( ) ( )1 1, F x F x G x G x khi x a th : ( )( )( )( )11lim limx a x aF x F xG x G x =nhl11:Nu( ) F x v( ) G x lhaiVCLkhix a ,( ) G x lVCLbcthphn ( ) F xth khix a ,( ) ( ) ( ) F x G x F x + (Quy tc ngt b cc VCL). V d 30. Tnh cc gii hn sau 3 53 223 5 10 3013 2 25 307 6) lim12 67 6 4 8) lim12 6 1000xxx x xax x xx x x x xbx x x x x ++ + + + + B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 17 ( )( )4 2 42) lim 3 1) lim 1xxc x x xd x x x+ + Gii. a) Ta c3 5 37 6 7 x x x x + khix 3 2 312 6 12 x x x x + khix . Vy3 5 33 3 27 6 7 7lim lim12 12 12 6x xx x x xx x x x += =+ . III. Hm s lin tc 1. nh ngha nh ngha 1. Cho f l mt hm s lin tc trong khong (a, b ), x0 l mt im thuc ( a, b). Ngi ta ni rng hm s f lin tc ti x0 nu: ( ) ( )00limx xf x f x= . (1) Nu hm s f khng lin tc ti x0, ta ni rng n gin on ti x0. Nu t:( ) ( )0 0,x x x y f x f x = + = , th ng thc (1) c th vit l: ( ) ( )00lim 0x xf x f x =( hay 0lim 0xy = . V d 31. Chng minh hm s 2y x =lin tc ti mi 0x . Ta c:x t( ) ( )2 22 2 20 0 0 0 0 0 0th ,2 x x x y x y x x x x x x x x = + = = = + = + ; 00 0 0 0lim 2 . lim lim . lim 0x x x xy x x x x = + = (pcm). V d 32. Chng minh hm ssin y x = lin tc ti mi 0x . Ta c: 0x , t( )0 0 0 0 0 0th sin , sin sin sin sin x x x y x y x x x x x = + = = = + = 02sin cos 2 sin2 2 2x x xx | |= + |\ . Do 0lim 0xy = . Tng t nh vy, c th chngminh c rngmi hm s s cp c bn u lin tc ti nhng im thuc min xc nh ca n. Nhnxt:ddngtrongtnhtanngitathngphtbiunhngha1di dng sau: i)f(x0) phi xc nh ii) 0x xlimf (x) phi tn ti iii) 00x xlimf (x) f (x )=V d 33. Xt s lin tc ca cc hm s sau B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 18 222a) f (x) x 2x 3x 9b) g(x)x 5x 6= = Gii. a) Ta c 2f (x) x 2x 3 = l mt hm s s cp nn xc nh, c gii hnfx D . Nn hm s lin tc ti mi x thuc tp xc nh Df b) Ta c22x 9g(x)x 5x 6= l mt phn thc hu t ( l mt dng ca hm s s cp) nnhmsxc nh, cgii hn fx D \ 2,3 = .Nnhmscnglintcti mi x thuc Df. Ring ti x=2, 3 ta nghi ng rng hm s c hoc khng lin tc nn ta lm nh sau: * Khi x= 3 th ta kim tra 3 iu kin ca hm lin tc: i) 22x 9 0g(x) g(3)x 5x 6 0= = khng xc nh nn ta c th b qua 2 iu kin kia v kt lun hm s khng lin tc ti x=3. * Tng t khi x=2. V d 34. Xt tnh lin tc ca hm sf (x) x = . nh ngha 2. (Lin tc tri, phi) * Lin tc tri Mt hm s f c gi l lin tc tri ti mt im x =c thuc Df nu tha mn 3 iu kin sau -f(c) c nh ngha ( xc nh). - x climf (x)phi tn ti. - x climf (x) f (c)= .Ta pht biu tng t cho trng hp lin tc phi. nh ngha 2. Hm s f c gi l lin tc trong khong m ( a, b) nu n lin tc ti mi im ca khong ; c gi l lin tc trong khong ng [a, b] nu n lin tc ti mi im ca khong m (a, b), lin tc phi ti a v lin tc tri ti b. V d 35. Tm tt c cc gi tr ca x m ti hm s f(x) khng lin tc B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 19 222x 1 x 1a) f (x) x 3x 4 1 x 35 x x 3xb) g(x)x 1x 3c) k(x)x 3x' < 1111= !1111+== 2. Cc php ton v hm s lin tc T cc nh l v gii hn ca tng, tch, thng v t nh ngha ca hm s lin tc ti mt im, c th d dng suy ra: nh l 12. Nu f v g l hai hm s lin tc ti x0 th: a)f + g lin tc ti x0. b)f.g lin tc ti x0. c) fg lin tc ti x0 nu( ) 0 g x . nhl13.Nuhms( ) u x = lintctix0,hms( ) y f u = lintcti ( )0 0u x =th hm s hp( )( ) ( ) y f g x f x = =( lin tc ti x0. V d 36. Xt tnh lin tc ca cc hm s sau: sinx, khi x 0xa)f (x)1 , khi x 01sin , khi x 0b)f (x)xa , khi x 0'1111=!11= 11+'111=!11=1+ ( )23x1 cosx, khi xx-c)f (x)1, khi x2ln(1 2x), khi x 01 ed)f (x)2, khi x 03'11111=!111=11+'11

11 1=!11111+ 2.1 Tnh cht ca hm s lin tc Cc nh l sau y nu ln nhng tnh cht c bn ca hm s lin tc. nhl 14. Nu hms( ) f xlin tc trn on [a, b] th n b chn trong on , tc l tn ti hai s m v M sao cho( ) [ ], m f x M x a b . nh l 15. Nu hm s( ) f xlin tc trn on [a, b] th n t gi tr nh nht m v gi tr ln nht M ca n trn on , tc l tn ti hai im [ ]1 2, , x x a b sao cho:( ) ( ) [ ]1 , ; f x m f x x a b = ( ) ( ) [ ]2 , f x M f x x a b = . B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 20 nh l 16. ( nh l v gi tr trung gian) Nu hm s( ) f xlin tc trn on [a, b], m v M l cc gi tr nh nht v ln nht can trn on th mi snm gia m v M, lun tn ti im[ ] , a b sao cho:( ) f = . Hqu.Nu( ) f x lintctrn[a,b],( ) ( ) . 0 f a f b < thtrongkhong(a,b)tnti mt imsao cho( ) 0 f = . Ch:Dngtnhchtcahmslintc,tachngminhccccngthcsau: ( )0ln 1lim 1+= ; 01lim 1e= ; 01lim lnaa= .Ttacthsuyrarngnu ( ) 0 x khix a th khix a :( ) ( ) ( ) ln 1 x x + ; ( )( ) 1xe x ; ( )( ) 1 lnxa x a . 2.2 Cc v d V d 37. Tnh 22 3lim4 2xxx++ Khix , cc t s v mu s u l cc VCL. Theo nguyn tc ngt b cc VCL 2 222 3 2lim lim lim .4 2 4 4x x xxx xx x x += =+ Vy22 3 2lim4 2 4xxx++=+, 22 3 2lim4 2 4xxx+= + V d 38. Tm 23lim 5xxx+. Ta c 2 2lim23 3lim 5 5 5 25xx xx xx + += = = V d 39. Tm 312 2lim .26 3xxx+ Ta phi kh dng v nh 00. t 326 x z + = , suy ra 326 x z = . Khi1 x th 327 z hay3 z . Ta c ( ) ( ) ( )( )( )3 3 23232 26 2 2 27 2 3 3 92 2 2 542 3 93 3 3 3 26 3z z z z zx zz zz z z z x + + = = = = = + + + khi3. z Vy ( )231 32 2lim lim2 3 9 5426 3x zxz zx = + + =+ V d 40. Tm 6sin6lim .3 2cos xxx| | |\ B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 21 t( )sin6,3 2cosxf xx | | |\ =c dng 00 khi 6x . t.6x z =Khi 6xth0 z . Ta c ( )22sin cossin sin2 23 3 cos sin3.2sin 2sin cos 3 2cos2 2 2 6z zz zf xz z zz zz= = = =| | ++ + |\ cos23sin cos2 2zz z=+ (khi0 z ). Vy( )06cos2lim lim 13sin cos2 2zxzf xz z = =+ V d 41. Tm 30sinlim .xtgx xx t( )3sin,tgx xf xx=ta c dng 00 khi0 x . Ta c ( )( )23 3 32sin .sinsin 1 cos sin sin cos2.cos cos cosxxx x x x xf xx x x x x x = = = khi 2220, sin , sin2 2 4x x xx x x| | = |\ Vy( )223 30 0 0 02sin .sin 2 .2 12 4lim lim lim lim .cos cos 4cos 2x x x xx xx xf xx x x x x = = = =V d 42. Tm( )130lim 1 .xxx+Ta gp dng1 khi0 x . Ta c( ) ( )111 133330 0lim 1 lim 1x xx xx x e e (+ = + = = ( . V d 43. Tm ( )0ln 1lim3 1xxx+ Ta phi kh dng v nh 00. Khi0 x th :( ) ln 1 x x + v3 1 ln3.xx Vy ( )0 0ln 1 1lim lim .3 1 ln3 ln3xx xx xx += = V d 44. Tm 21lim 1xxx| |+ |\ B mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 22 y, ta c dng v nh 1 khix . Ta c 2102 21 1lim 1 lim 1 1.x x xx xex x (| | | | ( + = + = = ||\ \ ( V d 45. Tnh cc gii hn sau: 22x 2x 2x 1xx2x 12xxx x2x 02x 1a) l im2x 3x 2x 1b) l im3x 35 4c) limx x 1

( ) 1

( ) l l l l 3. im gin on ca hm sHms( ) f x gilginonti 0x nunkhnglintcti 0x .Vy 0x l im gin on ca hm s( ) f xnu: -Hoc( ) f xkhng xc nh ti 0x ; -Hoc( ) f xxc nh ti 0x , nhng( ) ( )00limx xf x f x ; -Hoc khng tn ti( )0lim .x xf x Nu( ) f xkhng xc nh ti 0x , nhng( ) ( )0 0lim limx x x xf x f x + =th 0xgi l im gin on b c. Ch cn xc nh trn hm f ti 0x x =bng cch cho( )0f xbng gi tr chung ca hai gii hn trn,hm f tr thnh lin tc c ti 0x . Nu tn ti cc gii hn hu hn( ) ( )0 0lim , limx x x xf x f x + v( ) ( )0 0lim limx x x xf x f x + th 0xgi l im gin on loi 1. Hoc( ) ( )0 0lim limx x x xf x f x + gi l bc nhy ca f ti 0x . Nhng im gin on khng thuc loi 1 c gi l im gin on loi 2. V d 46. Hm s( )sin2xf x =khng xc nh ti x = 0, nhng 0sinlim 12xx= . Vy x = 0 l im gan on b c. Nu tab sung gi tr( ) 0 1 f = , th hm s tr nn lin tc c ti x = 0. V d 47.Hm s( )11xf xx+ = 00khixkhix> Xc nh ti mi x , nhng ( ) ( )0 0lim 1 1 limx xf x f x + = = . Vy x = 0 l im gin on loi 1, bc nhy ca hm f tiB mn Tan- Thng kKhoa Kinh T-Lut HQG Tp.HCM 23 x = 0 bng( ) 1 1 2 =V d 48. Hm s( )1f xx=khng xc nh ti x = 0. V 0 01 1lim , limx xx x+ = + = , im x = 0 l im gin on loi 2 V d 49. Tm im gin an ca cc hm s sau: 3 25 2xa) f (x)x x 4b) g(x) x 2x3x 1c) h(x)x 13 x 1= = == ( V d 50. Tm a, b cc hm s sau lin tc a) ax+1 x2f (x)sinx+b x2' 11111=!11

111+c) 2x x 1f (x)x ax+b |x|>1' 11=!11+ b) 3(x 1) x 0f (x) ax+b 0