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- 1. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 1 Gii hn dng v nh l nhng gii hn m ta khng th tm chng bng cch p dng trc tip cc nh l v gii hn v cc gii hn c bn trnh by trong Sch gio khoa. Do mun tnh gii hn dng v nh ca hm s, ta phi tm cch kh cc dng v nh bin i thnh dng xc nh ca gii hn Trong chng trnh ton THPT, cc dng v nh thng gp l : 0 , , , 0. , 1 0 Sau y l ni dung tng dng c th. I. GII HN DNG V NH 0 0 Gii hn dng v nh 0 0 l mt trong nhng gii hn thng gp nht i vi bi ton tnh gii hn ca hm s. tnh cc gii hn dng ny, phng php chung l s dng cc php bin i ( phn tch a thc thnh nhn t, nhn c t v mu vi biu thc lin hp, thm bt, ) kh cc thnh phn c gii hn bng 0, a v tnh gii hn xc nh. Chnh cc thnh phn c gii hn bng 0 ny gy nn dng v nh. tnh gii hn dng v nh 0 0 , trc ht gio vin cn rn luyn cho hc sinh k nng nhn dng. 1. Nhn dng gii hn v nh 0 0 gii bi ton tm gii hn ca hm s, hc sinh cn xc nh gii hn cn tm thuc dng xc nh hay v nh. Nu gii hn l v nh th phi xt xem n thuc dng v nh no c phng php gii thch hp. Bi vy vic rn luyn k nng nhn dng cho hc sinh c quan trng, gip hc sinh nh hng c cch gii, trnh nhng sai xt c th mc phi. i vi dng v nh 0 0 , vic nhn dng khng kh khn lm v hc sinh thng gp gii hn : 0x x f(x) lim g(x) m 0 0x x x x lim f(x) = lim g(x) = 0 WWW.MATHVN.COM www.MATHVN.com
2. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 2 Thc t hc sinh hay gp trng hp 0x x f(x) lim g(x) m 0 0f(x ) = (x ) = 0g . Ngoi ra trong mt s bi ton hc sinh phi thc hin cc php bin i chuyn v dng v nh 0 0 , sau mi p dng cc phng php kh cc thnh phn c gii hn bng 0. Khi ging dy, gio vin nn a ra mt s bi ton nhn mnh cho hc sinh vic nhn dng nh : 0x x f(x) lim g(x) m 0x x lim f(x) 0 hoc 0x x lim g(x) 0 Trnh tnh trng hc sinh khng nhn dng m p dng ngay phng php gii. V d p dng : (Yu cu chung ca nhng bi tp l : Tnh cc gii hn sau). V d 1 : 1 2x 2 x - 2 L = lim x +1 Bi gii : 1 2 2x 2 = x - 2 2 - 2 L = lim 0 x +1 2 1 V d 2 : 2 2x 1 - x + 2 L = lim x 1 Bi gii : 2 2x 1 - x + 2 L = lim = x 1 v 1 2 2 1 lim(x+2) = 1+2 = 3 lim(x - 1) = 1 - 1 = 0 x x V d 3 : 3 2x 1 1 3 L = lim x 1 x 1 Bi gii : 2 2 2x 1 x 1 x 1 x 1 = 1 3 x 3x +2 L = lim lim 3 x 1 x 1 x 1 (x-1)(x 2) (x-2) 1-2 1 lim lim (x 1)(x+1) (x+1) 1+1 2 WWW.MATHVN.COM www.MATHVN.com 3. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 3 Dng v nh 0 0 c nghin cu vi cc loi c th sau : 2. Loi 1 : 0x x f(x) lim g(x) m f(x), g(x) l cc a thc v f(x0) = g(x0) = 0 Phng php : Kh dng v nh bng cch phn tch c t v mu thnh nhn t vi nhn t chung l (x x0). Gi s : f(x) = (x x0).f1(x) v g(x) = (x x0).g1(x). Khi : 0 1 1 0 0 00 1 1 x x x x x x ) ) (x - x f (x) f (x)f(x) lim lim lim g(x) (x - x g (x) g (x) Nu gii hn 1 0 1 x x f (x) lim g (x) vn dng v nh 0 0 th ta lp li qu trnh kh n khi khng cn dng v nh. V d p dng : V d 4 : 2 4 2x 2 2x - 5x +2 L = lim x +x - 6 Bi gii : Ta phn tch c t v mu thnh nhn t vi nhn t chung : x - 2 2 4 2x 2 x 2 x 2 = 2x - 5x +2 (x - 2)(2x - 1) L = lim lim (x - 2)(x + 3)x +x - 6 2x - 1 2.2 1 3 lim x + 3 2 3 5 Vy 4 3 L 5 V d 5 : 2 5 x 2 2 x - 3x +2 L = lim - 4x + 4x Bi gii : 2 25 x 2 x 2 x 2 2 = x - 3x +2 (x - 2)(x - 1) L = lim lim (x - 2)- 4x + 4 x - 1 lim x - 2 x ( V gii hn ca t bng 1, gii hn ca mu bng 0) Vy 4L WWW.MATHVN.COM www.MATHVN.com 4. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 4 V d 6 : 2 2 3 n * 6 3 mx 1 + + x+x x +...+x - n L lim (m, n N ) x+x x +...+x - m Bi gii : Ta s phn tch t v mu thnh nhn t vi nhn t chung : x 1 bng cch tch v nhm nh sau : x + x2 + x3 + ... + xn n = (x 1) + (x2 1) + (x3 - 1) + ...+ (xn - 1) x + x2 + x3 + ... + xm m = (x 1) + (x2 1) + (x3 - 1) + ...+ (xm - 1) Khi : 22 2 2x 1 x 1 3 n3 n 6 3 m 3 m 1 - 1)+( - 1)+ + 1 - 1)+( - 1) lim lim (x- )+(x x +...+(x - 1)x+x x +...+x - nL x+x x +...+x - m (x- )+(x x +...+(x - 1) x 1 n-1 n-2 m-1 m-2 1 1 + (x + 1) +...+ ( ) 1 1 + (x + 1) +...+ ( ) lim (x- ) 1 (x- ) +1 x + x +...+ x + x + x +...+ x n-1 n-2 m-1 m-2x 1 1 + (x + 1) +...+ (x + x +...+ x +1) lim 1 + (x + 1) +...+ (x + x +...+ x +1) n-1 n-2 m-1 m-2 1 + (1 +1) +...+ (1 + 1 +...+ 1 +1) 1 + (1 +1) +...+ (1 + 1 +...+ 1 +1) n(n + 1) 1 2 3 ... n n(n + 1)2 m(m + 1)1 2 3 ... m m(m + 1) 2 Vy 6 n(n + 1) L m(m + 1) V d 7 : 4 3 2 7 4 3 21 2x - 5x +3x + x - 1 L lim 3x - 8x + 6x - 1x Bi gii : 3 2 7 3 2x 1 3 2 2 3 2 2 4 3 2 4 3 2 x 1 x 1 x 1 = (x-1)(2x - 3x +1) L = lim (x-1)(3x - 5x +x+1) 2x - 3x +1 (x-1)(2x - x -1) = = 3x - 5x + x +1 (x-1)(3x - 2x -1) 2x - 5x +3x + x - 1 lim 3x - 8x + 6x - 1 lim lim 2 2x 1 x 1 x 1 2x - x -1 (x -1)(2x+1) = lim = lim 3x - 2x -1 (x -1)(3x+1) 2x+1 2.1+1 3 = lim = = 3x+1 3.1+1 4 WWW.MATHVN.COM www.MATHVN.com 5. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 5 Vy 7 3 L = 4 Kt lun: Phng php gii bi tp loi ny l phn tch a thc thnh nhn t vi nhn t chung l x - x0. Yu cu i vi hc sinh l : Phi nm vng cc phng php phn tch a thc thnh nhn t, cc hng ng thc, cng thc phn tch tam thc bc hai, a thc bc ba thnh nhn t: 2 0 0 c f(x) = ax + bx + c = (x - x ) ax - x , ( f(x0) = 0) Ngoi cc hng ng thc ng nh, hc sinh cn nh cc hng ng thc b xung l : an - bn = (a - b)(an -1 + an - 2 b ++ abn - 2 + bn - 1 ), * n N an + bn = (a + b)(an -1 - an - 2 b +- abn - 2 + bn - 1 ), n l s t nhin l. hc sinh d nh, cn ly cc trng hp c th nh : n = 2, 3, 4 v trng hp c bit : xn - 1 = (x - 1)(xn - 1 + xn - 2 ++ x + 1). Tu theo c im tng bi m bin i mt cch linh hot kh dng v nh. Trong qu trnh thc hnh, nhiu khi sau cc bin i kh cc thnh phn c gii hn bng 0 ta vn gp gii hn dng v nh 0 0 mi ( thng l n gin hn so vi gii hn ban u). Ti y ta tip tc qu trnh kh n khi gii hn cn tm khng cn dng v nh 0 0 th thi. Bi tp t luyn 1) 3 4x 1 x 3x 2 lim x 4x 3 2) x 0 (1 x)(1 2x)(1 3x) 1 lim x 3) 100 50x 1 x 2x 1 lim x 2x 1 4) n 1 2x 1 x (n 1) n lim (x 1) 3. Loi 2 : 0x x f(x) lim g(x) m f(x), g(x) cha cc cn thc cng bc v f(x0)=g(x0)= 0 Phng php : Nhn c t v mu vi biu thc lin hp tng ng ca biu thc cha cn thc (gi tt l phng php nhn lin hp hay dng biu thc lin hp) trc cc nhn t x - x0 ra khi cc cn thc, nhm kh cc thnh phn c gii hn bng 0. Biu thc cha cn thc c th l t, mu hay c WWW.MATHVN.COM www.MATHVN.com 6. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 6 t v mu ca phn thc cn tm gii hn ). Lu l c th nhn lin hp mt hay nhiu ln kh dng v nh. Cc cng thc thng c s dng khi nhn lin hp l : 3 32 23 33 3 ( A B)( A B) = A - B , (A 0, B 0) ( A B)( A A B+ B ) =A B Gio vin cn cho hc sinh thy c hai cng thc ny xut pht t hai hng ng thc sau hc sinh d nh : 2 2 2 2 3 3 (a - b)(a + b) = a - b (a b)(a ab + b ) = a b V d p dng: V d 8 : 8 2x 2 3x - 2 - x L = lim x - 4 Bi gii : Nhn c t v mu vi biu thc lin hp tng ng, ta c : 8 2 2x 2 x 2 3x - 2 - x ( 3x - 2 - x)( 3x - 2 + x) L = lim lim x - 4 (x - 4)( 3x - 2 + x) 2 2x 2 x 2 x 2 3x - 2 - x (x - 2)(-x + 1) lim lim (x - 4)( 3x - 2 + x) (x - 2)(x + 2)( 3x - 2 + x) x + 1 2 + 1 1 lim 16(x + 2)( 3x - 2 + x) (2 + 2)( 3.2-2+2) Vy 8 1 L = 16 WWW.MATHVN.COM www.MATHVN.com 7. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 7 V d 9 : 9 1 x+2 1 L lim x+5 2 x Bi gii : 9 1 1 ( x+2 1)( x+2 1) ( x+5 2)x+2 1 L lim lim x+5 2 ( x+5 2)( x+5 2) ( x+2 1) x x 1 1 (x + 2 - 1)( x+5 2) (x + 1)( x+5 2) = lim lim (x + 5 - 4)( x+2 1) (x + 1)( x+2 1)x x 1 x+5 2 1 5 2 = lim 2 x+2 1 1 2 1x Vy L9 = 2 V d 10 : n * 10 m1 x - 1 L lim , (m, n N ) x - 1 x Bi gii : n 10 m1 n-1 n-2 m-1 m-2n n n n m m m m-1 m-2 n-1 n-2m m m m n n n1 x - 1 L lim x - 1 ( x - 1) ( x) +( x) +...+ x+1 ( x) +( x) +...+ x+1 =lim ( x - 1) ( x) +( x) +...+ x+1 ( x) +( x) +...+ x+1 x x m mm-1 m-2 m n n1 n-1 n-2 n (x - 1)( x + x +...+ x+1) = lim (x - 1)( x + x +...+ x+1) x m mm-1 m-2 m n n1 n-1 n-2 n x + x +...+ x+1 m = lim nx + x +...+ x+1 x Vy 10 m L = n Kt lun: Phng php dng biu thc lin hp l phng php ch yu c s dng tnh cc gii hn c cha cn thc cng bc. C th xem y l thut ton c bn cho php tnh c kh nhiu gii hn ca hm s cha cn thc, phng hng r rng, d hiu.Vic xc nh biu thc lin hp l khng qu WWW.MATHVN.COM www.MATHVN.com 8. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 8 kh khn i vi hc sinh. Tuy nhin gio vin cn rn luyn k nng xc nh v nhn biu thc lin hp khi tnh gii hn. Theo cch ny, nhiu bi ton tuy gii c nhng phi qua cc php bin i di dng vi biu thc cng knh. Nu dng cc gii khc nh thm bt, i bin s cho li gii ngn gn hn. Bi tp t luyn 1) 3 x 1 x x 3 lim x 1 2) 2 3x 2 x 4 lim 2 3x 2 3) 2 2x a x b a b lim x a 4) 3 23 2x 1 x 2 x x 1 lim x 1 5) n x 0 1 ax lim x 6) n n x 0 a x a lim x 4. Loi 3: 0x x f(x) lim g(x) m f(x) cha cc cn thc khng cng bc v f(x0)=g(x0)= 0 Phng php : S dng thut ton thm bt i vi f(x) c th nhn biu thc lin hp. Chng hn nh : 0 0 m n m n 0 0 0 x x x x u(x) v(x)f(x) L= lim = lim ,( u(x ) v(x ) = 0,g(x ) = 0) g(x) g(x) Ta bin i : 0 0 0 0 m nm n x x x x m n x x x x u(x) - c + c - v(x)u(x)- v(x) L lim lim g(x) g(x) u(x) - c v(x) - c = lim lim g(x) g(x) Ti y cc gii hn 0 0 m n 1 2x x x x u(x) - c v(x) - c L lim , L lim g(x) g(x) u tnh c bng cch nhn lin hp. V d p dng : V d 11 : 3 11 2x 1 x+3 x+7 L lim x 3x+2 WWW.MATHVN.COM www.MATHVN.com 9. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 9 Bi gii : x 1 x 1 x 1 x 1 3 3 11 2 2 3 2 2 lim lim lim lim x+3 x+7 ( x+3 2) + (2 x+7) L x 3x+2 x 3x+2 x+3 2 2 x+7 = x 3x+2 x 3x+2 23 3 3 2 2 23 3x 1 x 1 (2 x+7) 4 2 x+7 ( x+7)( x+3 2)( x+3+2) = lim lim (x 3x+2)( x+3+2) (x 3x+2) 4 2 x+7 ( x+7) 2 2 23 3x 1 x 1 x+3 4 8 (x+7) =lim lim (x 3x+2)( x+3+2) (x 3x+2) 4 2 x+7 ( x+7) x 1 x 1 23 3 x 1 1 x = lim lim (x 1)(x 2)( x+3+2) (x 1)(x 2) 4 2 x+7 ( x+7) x 1 x 1 23 3 1 1 = lim lim (x 2)( x+3+2) (x 2) 4 2 x+7 ( x+7) 23 3 1 1 = (1 2)( 1+3+2) (1 2) 4 2 1+7 ( 1+7) 1 1 1 = 4 12 6 Vy 11 1 L 6 V d 12 : 3 12 20 1+2x - 1+3x L lim xx Bi gii : 3 3 12 2 20 0 1+2x - (x+1) + (x+1) - 1+3x1+2x - 1+3x L lim lim x x x x 3 2 20 0 1+2x - (x+1) (x+1) - 1+3x =lim +lim x x x x WWW.MATHVN.COM www.MATHVN.com 10. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 10 0 2 2 23 3 3 0 2 2 23 3 1+2x - (x+1) 1+2x +(x+1) =lim x 1+2x +(x+1) (x+1) - 1+3x (x+1) ( 1) 1+3x ( 1+3x) +lim x (x+1) ( 1) 1+3x ( 1+3x) x x x x 2 3 2 2 2 23 30 0 2 23 30 0 (1+2x) - (x+1) (x+1) - (1+3x) lim lim x 1+2x +(x+1) x (x+1) (x 1) 1+3x ( 1+3x) - 1 x+3 lim lim 1+2x +(x+1) (x+1) (x 1) 1+3x ( 1+3x) x x x x 2 23 3 - 1 0+3 1+2.0 +(0+1) (0+1) (0 1) 1+3.0 ( 1+3.0) 1 1 1 2 2 Vy 12 1 L 2 Kt lun : Phng php chung tnh cc gii hn ca biu thc cha cc cn thc khng cng bc l thm, bt mt lng no , tch thnh nhiu gii hn ri nhn lin hp. Cn lu l c th thm bt mt hng s ( thng chn l u(x0) hoc v(x0)) hay mt biu thc. Vic thm bt da trn c im tng bi v phi tht tinh t. Thut ton thm bt cn c p dng hiu qu i vi cc dng v nh khc. Bi tp t luyn 1) 3 x 0 1 x 1 x lim x 2) 3 2x 2 x 11 8x 43 lim 2x 3x 2 3) n m x 0 1 ax 1 bx lim x 4) 3 2 x 0 2x 1 x 1 lim sin x 5) 3 4x 7 x 2 x 20 lim x 9 2 6) 3 2x 0 1 4x 1 6x lim x 5. Gii hn dng v nh 0 0 ca hm s lng gic WWW.MATHVN.COM www.MATHVN.com 11. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 11 Phng php : Thc hin cc php bin i i s v lng gic s dng cc kt qu gii hn c bn sau y : +) x 0 x 0 sinx x lim 1, lim 1 x sinx +) x 0 x 0 x 0 sinax sinax sinax lim lim( .a) =a.lim =a x ax ax +) x 0 x 0 x 0 x 0 x 0 sinax sinax bx ax sinax bx ax a lim lim( . . ) lim .lim .lim sinbx ax sinbx bx ax sinbx bx b +) x 0 x 0 x 0 x 0 tgax sinax a sinax a lim lim( . ) lim .lim a x ax cosax ax cosax Trong qu trnh bin i, hc sinh cn vn dng linh hot cc cng thc lng gic, thm bt, nhn lin hp V d p dng V d 13 : 13 x 0 1+sinax - cosax L lim 1- sinbx - cosbx Bi gii : 13 x 0 x 0 1+sinax - cosax 1- cosax+sinax L lim lim 1- sinbx - cosbx 1- cosbx - sinbx 2 x 0 x 02 ax ax axax ax ax 2sin sin cos2sin +2sin cos 2 2 22 2 2=lim lim bx bx bx bx bx bx2sin - 2sin cos 2sin sin - cos 2 2 2 2 2 2 x 0 x 0 ax ax ax sin sin cos a2 2 2=lim .lim bx bx bx bsin sin - cos 2 2 2 Vy 13 a L b V d 14 : 14 2x 0 1 cosax L lim x Bi gii : WWW.MATHVN.COM www.MATHVN.com 12. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 12 2 2 2 2 2 2 14 2 2x 0 x 0 x 0 x 0 ax ax ax 2sin sin sin 1 cosax a a a2 2 2L lim lim lim . lim ax axx x 2 2 2 2 2 Vy 2 14 a L 2 V d 15 : 15 20 1 xsinx - cos2x L lim sin xx Bi gii : 15 2 20 0 1 xsinx - cos2x (1 - cos2x) xsinx L lim lim sin x sin xx x 2 2 20 0 0 0 0 2 2sin x xsinx sinx(2sinx x) 2sinx x lim lim lim sin xsin x sin x x x lim 2 lim 2 1 3 sin x sin x x x x x x Vy L15 = 3 V d 16 : * 16 2x 0 1- cosx.cos2x...cosnx L lim (n N ) x Bi gii : 16 2x 0 2x 0 1- cosx.cos2x...cosnx L lim x 1-cosx+cosx-cosxcos2x+...+cosx.cos2x...cos(n-1)x-cosx.cos2x...cosnx lim x 2x 0 2 2 2x 0 x 0 x 0 ... 1-cosx+cosx(1- cos2x)+...+cosx.cos2x...cos(n-1)x(1- cosnx) lim x 1-cosx cosx(1-cos2x) cosx.cos2x...cos(n-1)x(1- cosnx) lim lim lim x x x Theo kt qu bi 14 ta c : 2 2x 0 1 2 1-cosx lim x 2 2 2x 0 x 0 x 0 . cosx(1-cos2x) 1-cos2x 2 lim lim cosx lim 2x x WWW.MATHVN.COM www.MATHVN.com 13. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 13 2x 0 2 2x 0 x 0 x 0 x 0 . ... . cosx.cos2x...cos(n-1)x(1- cosnx) lim x 1- cosnx n lim cosx lim cos2x lim cos(n-1)x lim 2x Do 2 2 2 2 2 2 16 1 2 n 1 2 ... n n(n+1)(2n+1) L ... 2 2 2 2 12 Trong bi tp ny ta s dng thut thm bt : cosx, cosxcos2x,, cosxcos2xcos(n - 1)x bin i v tnh gii hn cho. C th nhn thy thut thm bt ng vai tr quan trng trong k nng bin i i vi bi tp ny. V d 17 : 2 17 2x 0 1 x cosx L lim x Bi gii : 2 2 17 2 2x 0 x 0 (1 x cosx 1 x 1) (1 cosx) L lim lim x x 2 2 2 2 2 2 2x 0 x 0 x 0 x 02 2 x ( ( 2 ( 2sin1 x 1 1 cosx 1 x 1) 1 x 1) lim lim lim lim x x xx 1 x 1) 2 2 2 2x 0 x 0 x 0 x 02 2 2 x x 2 2 . ( 2sin sin1 x 1 1 1 lim lim lim lim x 2xx 1 x 1) 1 x 1 2 1 1 1 2 2 Vy L17 = 1. Kt lun : kh dng v nh i vi hm s lng gic, hc sinh cn nm vng v vn dng linh hot cc php bin i i s, lng gic cng nh p dng cc gii hn c bn. y ch c gii hn x 0 sinx lim 1 x c s dng trc tip, cc kt qu cn li khi lm bi phi chng minh li. WWW.MATHVN.COM www.MATHVN.com 14. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 14 vn dng gii hn x 0 sinx lim 1 x , cn a hm s cn tnh gii hn v dng : 0 0 0x x x x x x sinf(x) f(x) tgf(x) lim , lim , lim f(x) sinf(x) f(x) vi 0x x lim f(x) 0 bng cch thm, bt, i bin hay nhn, chia ng thi vi mt lng thch hp no . Trong khi gii bi tp, hc sinh c th gp kh khn, lng tng a v cc dng trn. Gio vin cn khc phc bng cch cho hc sinh lm cc bi tp nh : 2 20 1 sinx sin(x 1) lim , lim , ... 1 cosx x 3x+2x x Bi tp t luyn Tnh cc gii hn sau : 1) 0 1+sinx 1 sinx lim tgxx 2) 0 (a+x)sin(a+x) asina lim xx 3) x 0 1 cosxcos2xco3x lim 1 cosx 4) 2 20 2sin x+sinx 1 lim 2sin x 3sinx+1x 5) 3 3x 4 1 cotg x lim 2 cotgx cotg x 6) 3 x 0 1 cosx cos2x cos3x lim 1 cos2x 6. Gii hn dng v nh 0 0 ca hm s m v lgarit. Phng php : Thc hin cc php bin i v s dng cc gii hn c bn sau y : +) x x 0 1 lim 1 x e +) x 0 ln(1 x) lim 1 x Cc gii hn trn u c tha nhn hoc chng minh trong Sch gio khoa. Ngoi ra gio vin cn a ra cho hc sinh hai gii hn sau : WWW.MATHVN.COM www.MATHVN.com 15. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 15 +) x xlna x 0 x 0 a 1 1 lim lim .lna lna x x.lna e ( V xlna x 0 1 lim 1 xlna e ) +) x 0 x 0 x 0 a . log (1 x) ln(1 x) ln(1 x)1 lim lim lim lna x x.lna lna x V d p dng : V d 18 : ax bx 18 x 0 L lim x e e Bi gii : x 0 x 0 ax bxax bx 18 1) 1) lim lim ( ( L x x e ee e ax bx x 0 x 0 ax bx x 0 x 0 ( 1) ( 1) lim lim x x ( 1) ( 1) a. lim b. lim ax bx a b e e e e Vy L18 = a - b. Trong bi tp ny s dng gii hn c bn ta thc hin thm bt 1 v tch thnh hai gii hn. Cn nhn mnh cho hc sinh khi x 0 th ax 0 , do vy ax bx x 0 x 0 ( 1) ( 1) lim 1, lim 1 ax bx e e . V d 19 : sin2x sinx 19 x 0 L lim sinx e e Bi gii : sin2x sinxsin2x sinx 19 x 0 x 0 1) 1)( ( L lim lim sinx sinx e ee e sin2x sinx x 0 x 0 sin2x sinx x 0 x 0 1 1 lim lim sinx sinx 1 1 lim .2cosx lim sin2x sinx e e e e x 0 x 0 x 0 sin2x sinx . (2cosx) 1 1 lim lim lim sin2x sinx 2 1 1 e e WWW.MATHVN.COM www.MATHVN.com 16. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 16 Vy L19 = 1. V d 20 : x 2 20 x 2 2 x L lim x 2 Bi gii : x 2 x 2 20 x 2 x 2 4) 4)2 x (2 (x L lim lim x 2 x 2 x 2x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 1) 2)(x+2)4 4 1 4 4(2 (x2 x lim lim lim lim x 2 x 2 x 2 x 2 2 lim lim (x+2) 4ln2 4 x 2 Vy L20 = 4ln2 - 4 V d 21 : 223 2x 21 2x 0 1 x L lim ln(1+x ) e Bi gii : 22 33 2 2x2 2x 21 2 2x 0 x 0 ( 1)1 x 1) (1 x L lim lim ln(1+x ) ln(1+x ) ee 2 23 32 2x 2 2x 2 2 2x 0 x 0 x 0 ( 1) 11 x 1) ( 1 x 1 lim lim lim ln(1+x ) ln(1+x ) ln(1+x ) e e 3 32 23 32 2 23 2 2 2 2 2 2 x 0 x 02 2x( ( ) 1 ( ) 1 . 1 2x ln(1+x ) 1 x 1)( 1 x 1 x ) lim lim ( 1 x 1 x )ln(1+x ) 2x e 2 2 232 2 23 2 2 2x x 0 x 0 x 02 . ( ) 1 1 2x ln(1+x ) x lim lim lim 2x( 1 x 1 x )ln(1+x ) e 22 2 232 23 2 2 2x x 0 x 0 x 0 x 02 . . ( ) 1 x 1 ln(1+x ) 2x ln(1+x ) 1 7 .1 1.( 2) 3 3 1 lim lim lim lim 2x1 x 1 x e Vy 21 7 L 3 Kt lun : WWW.MATHVN.COM www.MATHVN.com 17. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 17 tnh cc gii hn dng v nh ca hm s m v lgarit, hc sinh thc hin cc php bin i p dng cc gii hn c bn. Yu cu hc sinh phi thnh tho cc php ton v lu tha v lgarit. s dng cc gii hn c bn, bng cch thm, bt, nhn lin hp, hc sinh phi bin i hm s cn tm gii hn v mt trong cc dng : 0 0 0 0 f(x) f(x) a x x x x x x x x ln 1+f(x) log 1+f(x)1 a 1 lim , lim , lim , lim f(x) f(x) f(x) f(x) e vi 0x x lim f(x) 0 Bi tp t luyn Tnh cc gii hn sau : 1) 2) x x x xx 0 5 4 3 lim 9 3) x 0 2 2 x3 cosx x lim 4) x 3 4x 0 (1 )(1 cosx) lim 2x 3x e 5) x 0 1 1 x lim .ln x 1 x 6) sin2x sinx 2x 0 lim 5x + tg x e e II. GII HN DNG V NH Gii hn dng v nh c dng l : 0x x (x ) f(x) L lim g(x) trong : 0 0x x x x (x ) (x ) f(x) g(x)lim lim kh dng v nh ny, phng php thng thng l chia c t v mu cho lu tha bc cao nht ca t v mu ca phn thc f(x) g(x) . C th nh sau : 1) Nu f(x), g(x) l cc a thc c bc tng ng l m, n th ta chia c f(x), g(x) cho xk vi k = max{m, n} m m 1 m m 1 1 0 n n 1x n n 1 1 0 a x +a x +...+a x+a L lim b x +b x +...+b x+b vi * m na ,b 0, m,n N Khi xy ra mt trong ba trng hp sau : WWW.MATHVN.COM www.MATHVN.com 18. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 18 +) m = n (bc ca t v mu bng nhau), chia c t v mu cho xn ta c: 0m 1 1 nn 1 m m 0n 1 1 n n nn 1 x x m n aa a a ax xx bb b b b x xx a lim lim b + +...+ + L + +...+ + +) m > n (bc ca t ln hn bc ca mu, k = m), chia c t v mu cho xm ta c : n m nm n+1 n m n 0m 1 1 mm 1 m x x 0n 1 1 m m b x b x aa a a ax xxlim lim bb b x x x + +...+ + L +...+ ++ +) m < n (bc ca t nh hn bc ca mu, k = n), tng t nh trn ta c : 0m m 1 n m nn m+1 x 0n 1 n n aaa ... x xxlim 0 bb b ... x x L Hc sinh cn vn dng kt qu : 0 0 0 0x x x x x x x x 1 1 lim f(x) lim 0, lim f(x) 0 lim f(x) f(x) Sau khi xt ba trng hp ny, hc sinh cn t rt ra nhn xt kt qu gii hn cn tm da vo bc ca t v mu. Lu l c th chia t v mu cho xh vi h min{m, n}. 2) Nu f(x), g(x) l cc biu thc c cha cn thc th ta quy c ly gi tr m k ( trong k l bc ca cn thc, m l s m cao nht ca cc s hng trong cn thc) l bc ca cn thc . Bc ca t ( mu) c xc nh l bc cao nht cc biu thc trn t ( di mu). Sau ta p dng phng php kh nh vi trng hp f(x), g(x) l cc a thc. Qua hc sinh c th d dng phn on kt qu gii hn dng cn tm. V d p dng : V d 22 : 3 2 22 3x 2x 3x 1 L lim 5x 6 WWW.MATHVN.COM www.MATHVN.com 19. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 19 Bi gii : Chia c t v mu cho x3 ta c : 3 3 3 2 22 3x x 3 1 2 2x x 6 55 x 2x 3x 1 L lim lim 5x 6 Vy 22 2 L 5 . Ta c th trnh by theo cch sau : 3 3 3 3 33 3 2 22 3x x x 3 1 3 1x 2 2x x 2x x 6 56 5x 5 xx 2x 3x 1 L lim lim lim 5x 6 V d 23 : 2 2 2x23 3x (2x 1)(3x x+2) 2x+1 4x limL Bi gii : 2 2 4 2 2 2x x23 3x (2x 1)(3x x+2) 12x (2x+1)(3x x+2) 2x+1 4x 4x (2x+1) lim limL 3 2 3 2 2 3 x x 5x x+2 4x 5 1 2 4 4 1x x x 4 8 28+ x 4x lim lim 8x Vy 23 1 2 L V d 24 : 24 5x (x 1)(x 2)(x 3)(x 4)(x 5) L lim (5x 1) Bi gii : 24 5x (x 1)(x 2)(x 3)(x 4)(x 5) L lim (5x 1) 5 5x 1 2 3 4 5 1 1 1 1 1 x x x x x 1 51 5 x lim Vy 24 5 1 L 5 WWW.MATHVN.COM www.MATHVN.com 20. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 20 V d 25 : 25 x 2 x+3 L lim x 1 Bi gii : Chia c t v mu cho x ta c : 25 x x2 2 3 x+3 xL lim lim x 1 x 1 x 1+ V phi a x vo trong cn bc hai nn ta xt hai trng hp : *) 2 x x > 0 x x Khi : x + x + x +2 2 2 2 1 3 3 3 x x xlim lim lim 1 x 1 x 1 x x x 1+ 1+ 1+ 1 *) 2 x x < 0 x x Khi , ta c : x x x2 2 2 2 1 3 3 3 x x xlim lim lim 1 x 1 x 1 xx x 1+ 1+ 1+ 1 V x x2 2 1, 1lim lim x+3 x+3 x 1 x 1 nn khng tn ti x 2 x+3 lim x 1 V d 26 : 32 2 26 5x 4 4 4 9x 1 x 4 L lim 16x 3 x 7 Bi gii : Chia c t v mu cho x ta c : 32 2 32 2 26 5 5x x4 44 4 4 4 9x 1 x 4 9x 1 x 4 x xL lim lim 16x 3 x 7 16x 3 x 7 x x 2 3 3 x 4 4 5 5 9x 1 1 4 x x xlim 16x 3 1 7 x x x Tng t Bi 25, ta xt hai trng hp : WWW.MATHVN.COM www.MATHVN.com 21. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 21 *) 4 42, xx x > 0 x x x Khi : 2 3 3 2 3 + 26 4x x4 4 5 545 3 54 9x 1 1 4 1 9 9 0 3x x x x xL lim lim 216 0316x 3 1 7 xx xxx 1 4 x 1 716 x *) 4 42 , xx x < 0 x x x Khi ta c : 22 4 4 4 22 33 3 33 3 26 x x x4 44 4 4 55 5 55 5 1 xx 3 xx 9x 1 1 49x 1 1 4 1 4 9 x xx x xx xL lim lim lim 1 716x 3 1 7 16x 3 1 7 16 x xx x xx x 2 4 3 3 4 4 5 5 x 1 x 3 x 1 4 9 9 0 3x x 21 7 16 0 16 x x lim V + 26 26L L nn ta c : 26 3 L 2 Kt lun : So vi dng v nh 0 0 , dng v nh d tm hn. Hc sinh cn xc nh ng dng v ch cn quan tm n bc ca t v mu t phn on kt qu gii hn cn tm. Ch i vi gii hn dng ca hm s c cha cn thc ta khng nhn lin hp. y l im khc bit cn phn bit trnh nhm ln. Vi gii hn khi x , cn lu hai kh nng x v x trong php ly gii hn c cha cn bc chn. Nu hc sinh khng n vn ny th rt d mc phi sai lm. Hn na trng hp ny cn lin quan ti bi ton tm tim cn ca hm s cha cn thc. Bi tp t luyn 1) 2 3 2 2x 2x 3 4x+7 lim 3x 1 10x 9 2) 20 30 50x (2x 3) (3x+2) lim (2x+1) WWW.MATHVN.COM www.MATHVN.com 22. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 22 3) 2 n n+1x n 2 (x+1)(x 1)...(x 1) lim (nx) 1 4) 2 x 2 x 2x 3x lim 4x 4 x+2 5) 34 5 2 4x 4 3 x 1 x 2 lim x 1 x 2 6) 3 3 4x ln(1 x x) lim ln(1 x x) III. GII HN DNG V NH Dng tng qut ca gii hn ny l : 0x x (x ) lim f(x) g(x) trong 0 0x x x x (x ) (x ) lim f(x) lim f(x) Phng php ch yu kh dng v nh ny l bin i chng v dng v nh 0 , 0 bng cch i bin, nhn lin hp, thm bt, V d p dng : V d 27 : 2 27 x L lim x x x Bi gii : Nhn v chia biu thc lin hp tng ng l : 2 x x+x , ta c : 2 2 2 27 x 2x ( x x x)( x x+x) L lim x x x x x+x lim 2 2 2 2x x x x x+x x x+x x x x lim lim V x nn chia c t v mu cho x ta c : 2x x 1 1 21x x+x 1 1 x x lim lim Vy 27 1 L 2 Trong v d ny, bng cch nhn lin hp, ta chuyn gii hn cn tm t dng sang dng . http://kinhhoa.violet.vnWWW.MATHVN.COM www.MATHVN.com 23. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 23 V d 28 : 28 x L lim x+ x x Bi gii : 28 x x ( x+ x x)( x+ x x) L lim x+ x x lim x+ x x x x x+ x x x lim lim x+ x x x+ x x x x 1 x 1 1 lim lim 2x+ x x 1+ 1 x ( chia c t v mu cho x ) Vy 28 1 L 2 V d 29 : 2 29 x L lim x x 3 x Bi gii : Trong v d ny cn lu khi x cn xt hai trng hp x v x +) Khi x th : 2 2 x x 3 xx x 3 Do 2 x lim x x 3 x +) Khi x th gii hn c dng . Ta kh bng cch nhn lin hp bnh thng 2 2 2 x x 2 ( x x 3 x)( x x 3 x) lim x x 3 x lim x x 3 x 2 2 x x x2 2 2 3 1x x 3 x x 3 lim lim lim x x 3 x x x 3 x x x 3 1 x x Khi x th x < 0, do 2 x x WWW.MATHVN.COM www.MATHVN.com 24. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 24 x x2 2 3 3 1 1 1 1x xlim lim 2 21 3x x 3 1 11 x xx Vy 2 x lim x x 3 x , 2 x 1 lim x x 3 x 2 Qua v d ny mt ln na nhn mnh cho hc sinh ch vi gii hn khi x cn xt x v x i vi hm s cha cn thc bc chn. V d 30 : 3 3 2 2 30 x L lim x 3x x 2x Bi gii : V hm s cn tm gii hn cha cc cn thc khng cng bc nn ta thm bt c th nhn lin hp. 3 33 2 2 3 2 2 30 x x )L lim x 3x x 2x lim ( x 3x x ( x 2x x) x x 3 2 23 1 2lim limx 3x x x 2x x G G +) 3 2 3 21 3 2 3 2 23 3 3 23 x x x 3x G x 3x x x 3x x x lim x 3x x lim 2 2 3 2 3 2 23 3 3 3 x x 3 3 3 1 33 3x 3x x x 3x x 1 1 1 x xx lim lim +) 2 2 2 2 x x2 x 2x x x 2x x G lim x 2x x lim 2x x 2 2 2 1 22x 2x x 1 1 xx lim lim Vy L30 = G1 - G2 = 2 V d 31 : * 31 m nx 1 m n L lim , (m, n N ) 1 x 1 x Bi gii : WWW.MATHVN.COM www.MATHVN.com 25. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 25 31 m n m nx 1 x 1 m n m 1 n 1 L lim lim 1 x 1 x 1 x 1 x 1 x 1 x 1 2m nx 1 x 1 m 1 n 1 lim lim G G 1 x 1 x 1 x 1 x +) 2 m 1 1 m mx 1 x 1 m 1 m (1 x x ... x ) G lim lim 1 x 1 x 1 x 2 m 1 mx 1 (1 x) (1 x ) ... (1 x ) lim 1 x m 2 m 1x 1 (1 x) 1 (1 x) ... (1 x ... x ) lim (1 x)(1 x ... x ) m 2 m 1x 1 1 (1 x) ... (1 x ... x ) 1 2 ... m 1 m 1 lim 1 x ... x m 2 Tng t ta tnh c 2 n 1 G 2 Vy 31 1 2 m 1 n 1 m n L G G 2 2 2 Trong bi tp ny ta s dng thut ton thm, bt tch gii hn cn tm thnh hai gii hn v tnh cc gii hn ny bng cch bin i v dng 0 0 . Vic thm bt biu thc phi tinh tv ph thuc vo c im tng bi. Kt lun : i vi dng v nh , ta phi tu vo c im tng bi m vn dng linh hot cc k nng thm bt, nhn lin hp, phn tch thnh nhn t bin i v kh dng v nh. Ta thng chuyn chng v cc dng v nh d tnh hn l 0 0 , . Bi tp t luyn 1) x lim x x x x 2) 2 23 3 x lim (x 1) (x 1) 3) x lim x x x x x x 4) 3 2 x lim x 1 x WWW.MATHVN.COM www.MATHVN.com 26. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 26 5) x lim ln(5x 8) ln(3x 5) 6) 5 x lim (x 1)(x 2)...(x 5) x IV. GII HN DNG V NH 0. Dng tng qut ca gii hn ny l : 0 x (x x ) lim f(x).g(x) trong 0 0 x x (x x ) (x x ) lim f(x) 0, lim g(x) kh dng v nh ny, ta thng tm cch chuyn chng v dng gii hn khc d tnh hn nh 0 0 , bng cch nhn lin hp, thm bt, i bin V d p dng : V d 32 : 2 32 x L lim x x 5 x Bi gii : Ta kh dng v nh ny bng cch nhn lin hp a v dng v nh 2 2 2 32 2x x x( x 5 x)( x 5 x) L lim x x 5 x lim x 5 x WWW.MATHVN.COM www.MATHVN.com 27. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 27 2 2 2 2 2x x x x(x 5 x ) 5x 5 lim lim lim x 5 x x 5 x x 5 x x 2 x x > 0 x x Do : 2x x 2 5 5 5 lim lim 25x 5 x 1 1 xx Vy 32 5 L 2 V d 33 : 33 x 1 x L lim(1 x)tg 2 Bi gii : t t 1 x ta c : x 1 t 0 33 t 0 t 0 (1 t) t L lim t.tg lim t.tg 2 2 2 t 0 t 0 t 0 t t t 2 22lim t.cotg lim lim t t2 tg tg 2 2 Vy 33 2 L Bi tp t luyn 1) 2 x lim x 4x 9 2x 2) 2 4 4 x lim x 3x 5 3x 2 3) 2 33 x lim x 4x 5 8x 1 4) x 4 lim tg2x.tg x 4 5) 2 2 x a x lim a x tg 2a 6) 1 1 2 x x x lim x e e 2 V. GII HN DNG V NH 1 Dng tng qut ca gii hn ny l : WWW.MATHVN.COM www.MATHVN.com 28. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 28 0 g(x) x x lim f(x) , trong 0 0x x x x lim f(x) 1, lim g(x) Hai gii hn c bn thng c s dng khi tnh gii hn dng v nh 1 l : +) x x 1 lim 1 x e (1) +) 1 x x lim 1 x e (2) Trong qu trnh vn dng, hc sinh bin i v dng 0x x f(x) 1 lim 1 f(x) e nu 0x x lim f(x) 0 1 g(x) x x lim 1 g(x) e nu 0x x lim g(x) 0 bin i gii hn cn tm, hc sinh vn dng mnh sau (da vo tnh lin tc ca hm s m). Nu hai hm s f(x), g(x) tho mn cc iu kin : 1) 0x x lim f(x) a 0 2) 0x x lim g(x) b th 0 g(x) b x x lim f(x) a Hai gii hn c bn v mnh trn l c s tnh cc gii hn dng v nh 1 V d p dng V d 34 : 1 x 34 x 0 L lim 1+ sin2x Bi gii : sin 2x 1 1 sin 2x 1 x. x sin 2x x sin 2x 34 x 0 x 0 x 0 L lim 1+ sin2x lim 1+ sin2x lim 1+ sin2x WWW.MATHVN.COM www.MATHVN.com 29. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 29 Ta c : 1 sin 2x x 0 lim 1+ sin2x e ( hc sinh d hiu nn t t = sin2x) x 0 x 0 sin2x sin2x lim 2lim 0 x 2x Do : sin 2x 1 x 2 sin 2x 34 x 0 L lim 1+ sin2x e V d 5 : 4 3x 35 x x 1 L lim x 2 Bi gii : s dng gii hn c bn ta bin i : x 1 1 1 x 2 (x 2) 4 3x (x 2).4 3x (x 2) 35 x x x 1 1 L lim lim 1 x 2 (x 2) V (x 2) x x x x 1 lim 1 e (x 2) 4 34 3x 3x 4 xlim lim lim 3 2(x 2) x 2 1 x nn 3 35L e Bi 36 : tg2 y 4 36 t 0 L lim tg y 4 Bi gii : t y x , x y 0 4 4 .Ta c : 2 1 tg y tg2 y 2tgy4 36 t 0 t 0 1 tgy L lim tg y lim 4 1 tgy 2 2 2tgy 1 tg y .1 tg y 1 tgy 1 tgy 2tgy 2tgy 2tgy t 0 t 0 2tgy 2tgy lim 1 lim 1 1 tgy 1 tgy WWW.MATHVN.COM www.MATHVN.com 30. Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 30 V 1 tgy 2tgy y 0 2tgy lim 1 e 1 tgy v 2 y 0 y 0 2tgy 1 tg y . 1 tgy 1 1 tgy 2tgy lim lim nn 1 36L e Kt lun : Vi dng v nh 1 , vic nhn dng khng kh khn i vi hc sinh. Tuy nhin, lm c bi tp, hc sinh phi vn dng tt cc k nng a cc gii hn cn tm v mt trong hai gii hn c bn (1) v (2). Hai k nng ch yu c s dng l i bin v thm bt. Bi tp t luyn 1) 2 cot g x2 x 0 lim 1 x 2) 1 sin x x 0 1 tgx lim 1 sin x 3) 2 x2 2x x 3 lim x 2 3) cot g x x 1 lim 1 sin x 5) 2 1 x x 0 lim(cos2x) 6) x x 1 1 lim sin cos x x WWW.MATHVN.COM www.MATHVN.com