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. ,10 Sau y l ni dung tng dng c th.I. GII HN DNG V NH 00

Giihndngvnh 00lmttrongnhnggiihnthnggpnht ivibitontnhgiihncahms.tnhccgiihndngny, phng php chung l s dng cc php bin i ( phn tch a thc thnh nhn t,nhnctvmuvibiuthclinhp,thmbt,)khccthnh 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 nh00, trc ht gio vin cn rn luyn cho hc sinh k nng nhn dng. 1.Nhn dng gii hn v nh 00 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 rnluynknngnhndngchohcsinhcquantrng,giphcsinhnh hng c cch gii, trnh nhng sai xt c th mc phi. i vi dng v nh 00, vic nhn dng khng kh khn lm v hc sinh thng gp gii hn : 0x xf(x)limg(x)m 0 0x x x xlimf(x) =limg(x) = 0 www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...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 xf(x)limg(x) m 0 0f(x ) =(x ) = 0 g . Ngoi ra trong mt s bi ton hc sinh phi thc hin cc php bin i chuyn v dng v nh 00, sau mi p dng cc phng phpkh 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 xf(x)limg(x)m 0x xlimf(x) 0= hoc 0x xlimg(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 2x - 2L=limx +1 Bi gii : 1 2 2x 2 = x - 2 2 - 2L=lim 0x +1 2 1=+ V d 2 : 2 2x1- x + 2L=limx 1 Bi gii : 2 2x 1 - x + 2L=lim = x 1 v 12 2 1lim(x+2) = 1+2 = 3lim(x - 1) = 1- 1 = 0xx V d 3 : 3 2x11 3L=limx 1 x 1 | | |\ . Bi gii : 22 2x 1 x 1x 1 x 1=1 3 x 3x +2L=lim lim3x 1 x 1 x 1(x-1)(x 2) (x-2) 1-2 1lim lim(x 1)(x+1) (x+1) 1+1 2 | | | | = = ||\ .\ .| |= = = |\ . www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 3 Dng v nh 00 c nghin cu vi cc loi c th sau : 2. Loi 1 : 0x xf(x)limg(x) m f(x), g(x) l cc a thc v f(x0) = g(x0) = 0Phng 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 10 0 00 1 1x x x x x x))(x - x f (x) f (x) f(x)lim lim limg(x) (x - x g (x) g (x) = =Nu gii hn 101x xf (x)limg (x) vn dng v nh00 th ta lp li qu trnh kh n khi khng cn dng v nh. V d p dng : V d 4 : 24 2x 22x - 5x +2L=limx +x - 6 Bi gii : Ta phn tch c t v mu thnh nhn t vi nhn t chung : x - 2 24 2x 2 x 2x 2 =2x - 5x +2 (x - 2)(2x - 1)L=lim lim(x - 2)(x + 3) x +x - 62x - 1 2.2 1 3 limx + 3 2 3 5 = == =+ Vy 43L5=V d 5 : 25x 2 2x - 3x +2L=lim- 4x + 4 x Bi gii : 22 5x 2 x 2x 22= x - 3x +2 (x - 2)(x - 1)L=lim lim(x - 2)- 4x + 4x - 1limx - 2x = == ( V gii hn ca t bng 1, gii hn ca mu bng 0) Vy 4L = www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 4 V d 6 : 223 n*6 3 mx 1+ +x+x x +...+x - nLlim (m, nN )x+x x +...+x - m= e 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 : 2 22 2x 1 x 13 n 3 n6 3 m 3 m1 - 1)+( - 1) ++ 1 - 1)+( - 1)limlim(x-)+(x x +...+(x - 1) x+x x +...+x - nLx+x x +...+x - m (x-)+(x x +...+(x - 1) = = =x 1n-1 n-2m-1 m-21 1 + (x + 1) +...+ ( )

1 1 + (x + 1) +...+ ( ) lim(x-) 1(x-) +1x + x +...+ x +x + x +...+ x ( ( = ( ( =n-1 n-2m-1 m-2x 1 1 + (x + 1) +...+ (x + x +...+ x +1) lim1 + (x + 1) +...+ (x + x +...+ x +1)= = n-1 n-2m-1 m-21 + (1 +1) +...+ (1 + 1 +...+ 1 +1) 1 + (1 +1) +...+ (1 + 1 +...+ 1 +1) = =n(n + 1)1 2 3 ... n n(n + 1)2m(m + 1)1 2 3 ... m m(m + 1)2+ + + += = =+ + + + Vy 6n(n + 1)Lm(m + 1)=V d 7 : 4 3 27 4 3 212x - 5x +3x + x - 1L lim3x - 8x + 6x - 1x=Bi gii :3 273 2x13 2 23 2 24 3 24 3 2x1x1 x1 =(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 - 1lim3x - 8x + 6x - 1lim lim 22x1 x1x12x - x -1 (x -1)(2x+1)=lim =lim3x - 2x -1 (x -1)(3x+1)2x+1 2.1+1 3=lim = =3x+1 3.1+1 4 = www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 5 Vy 73L =4 Kt lun: Phng php gii bi tp loi ny l phn tch a thc thnh nhn t vi nhn t chung lx - 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: 200cf(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 - 2b ++ abn - 2+ bn - 1), *n N ean + bn = (a + b)(an -1- an - 2b +- abn - 2+ bn - 1), n l s t nhin l. hcsinhdnh,cnlycctrnghpcthnh:n=2,3,4v 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 bng0ta vn gpgii hn dng vnh 00mi( 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 00 th thi. Bi tp t luyn 1) 34x 1x 3x 2limx 4x 3 + + 2) x 0(1 x)(1 2x)(1 3x) 1limx+ + + 3) 10050x 1x 2x 1limx 2x 1 + +4) n 12x 1x (n 1) nlim(x 1)+ + + 3. Loi 2 : 0x xf(x)limg(x) m f(x), g(x) cha cc cn thc cng bc v f(x0)=g(x0)= 0Phng 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 haydng biu thc lin hp) trc cc nhn tx - 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.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...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 3 2 2 3 3 3 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 22 2 3 3(a - b)(a + b) = a - b(a b)(aab + b ) = a b V d p dng:V d 8 : 82x23x - 2 - xL =limx - 4 Bi gii : Nhn c t v mu vi biu thc lin hp tng ng, ta c : 822x2 x23x - 2 - x ( 3x - 2 - x)( 3x - 2 + x)L =lim limx - 4(x - 4)( 3x - 2 + x) =

22x2 x2x23x - 2 - x (x - 2)(-x + 1)lim lim(x - 4)( 3x - 2 + x) (x - 2)(x + 2)( 3x - 2+ x) x + 1 2 + 1 1lim16(x + 2)( 3x - 2 + x) (2 + 2)( 3.2-2+2) = = = = = = Vy81L =16www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 7 V d 9 : 91x+2 1L limx+5 2 =x Bi gii :91 1( x+2 1)( x+2 1) ( x+5 2)x+2 1L lim limx+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 + += =+ + 1x+5 2 1 5 2= lim 2x+2 1 1 2 1x + + += =+ ++ Vy L9 = 2 V d 10 : n*10m1x - 1L lim ,(m, n N )x - 1= ex Bi gii :n10m1n-1 n-2 m-1 m-2 n n n n m m mm-1 m-2 n-1 n-2 m m m m n n n 1x- 1L limx- 1( x- 1) ( x) +( x) +...+ x+1 ( x) +( x) +...+ x+1=lim( x- 1) ( x) +( x) +...+ x+1 ( x) +( x) +...+ x+1= = (( = (( xx

m m m-1 m-2 mn n 1 n-1 n-2 n(x - 1)( x + x +...+ x+1)=lim(x - 1)( x + x +...+ x+1)

=x

m m m-1 m-2 mn n 1 n-1 n-2 nx + x +...+ x+1 m=limnx + x +...+ x+1=x Vy10mL= n Kt lun:Phngphpdngbiuthclinhplphngphpchyucs 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, phnghngrrng,dhiu.Vicxcnhbiuthclinhplkhngqu www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...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 cngknh. Nu dng cc gii khc nh thm bt, i bin s cho li gii ngn gn hn. Bi tp t luyn1) 3x 1x x 3limx 1 + 2) 23x 2x 4lim2 3x 2+ 3) 2 2x ax b a blimx a 4) 3 2 32x 1x 2 x x 1limx 1 + 5) nx 01 axlimx+6) n nx 0a x alimx+ 4. Loi 3:0x xf(x)limg(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 0m nm n0 0 0xx xxu(x) v(x) f(x)L= lim = lim ,( u(x ) v(x )= 0,g(x ) = 0)g(x) g(x) Ta bin i :0 00 0m nm nxx xxm nxx xxu(x)- c+c -v(x)u(x)- v(x)L lim limg(x) g(x)u(x)- c v(x)- c = lim limg(x) g(x) (( = = = Ti y cc gii hn 0 0m n1 2xx xxu(x) - c v(x) - cL lim ,L limg(x) g(x) = =u tnh c bng cch nhn lin hp. V d p dng : V d 11 : 311 2x1x+3x+7L limx3x+2= www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 9 Bi gii : x1 x1x1 x13 3112 232 2lim limlim limx+3x+7 ( x+3 2) + (2 x+7)Lx3x+2 x3x+2x+3 2 2 x+7=x3x+2 x3x+2

= = = + =

2 3 3 322 2 3 3 x1 x1 (2 x+7) 4 2 x+7 ( x+7)( x+3 2)( x+3+2)=lim lim(x3x+2)( x+3+2)(x3x+2) 4 2 x+7 ( x+7) ( + + + ( + + 22 2 3 3 x1 x1x+3 4 8 (x+7)=lim lim(x3x+2)( x+3+2)(x3x+2) 4 2 x+7 ( x+7) + = ( + +

x1 x1 2 3 3x 1 1 x=lim lim(x 1)(x 2)( x+3+2)(x 1)(x 2) 4 2 x+7 ( x+7) ( + = + + x1 x1 2 3 31 1=lim lim(x 2)( x+3+2)(x 2) 4 2 x+7 ( x+7) ( + = + +

2 3 31 1=(1 2)( 1+3+2)(1 2) 4 2 1+7 ( 1+7)1 1 1=4 12 6 ( + = + + + = Vy 111L6= V d 12 : 312201+2x -1+3xL limxx= Bi gii : 33122 20 01+2x - (x+1) + (x+1) -1+3x1+2x -1+3xL lim limx x (( = = =x x

32 20 01+2x - (x+1) (x+1) -1+3x=lim +limx x =x x www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 10 0 22 2 3 3 30 2 2 2 3 31+2x - (x+1) 1+2x +(x+1)=limx 1+2x +(x+1)(x+1) -1+3x (x+1) ( 1) 1+3x ( 1+3x)+limx (x+1) ( 1) 1+3x ( 1+3x) (( ( (( ( ++ + ++ + +xxxx 2 32 2 2 2 3 3 0 02 2 3 30 0(1+2x) - (x+1) (x+1)- (1+3x)lim limx 1+2x+(x+1) x (x+1) (x 1) 1+3x ( 1+3x)- 1 x+3lim lim1+2x+(x+1) (x+1) (x 1) 1+3x ( 1+3x)x xx x = + = ((+ + + = + =+ + +

2 2 3 3- 1 0+31+2.0 +(0+1) (0+1) (0 1) 1+3.0 ( 1+3.0)1 112 2= + =+ + += + = Vy 121L2= Kt lun : Phng php chung tnh cc gii hn ca biu thc cha cc cn thc khngcngbclthm,btmtlngno,tchthnhnhiugiihnri nhn lin hp. Cn lu lc th thm bt mt hng s ( thng chn l u(x0) hocv(x0))haymtbiuthc.Victhmbtdatrncimtngbiv phithttinht.Thuttonthmbtcncpdnghiuquivicc dng v nh khc. Bi tp t luyn1) 3x 01 x 1 xlimx+ 2) 32x 2x 11 8x 43lim2x 3x 2+ ++ 3) n mx 01 ax 1 bxlimx+ +4) 3 2x 02x 1 x 1limsinx+ + 5) 34x 7x 2 x 20limx 9 2+ ++ 6) 32x 01 4x 1 6xlimx+ + 5. Gii hn dng v nh 00 ca hm s lng gic www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...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 sinaxsinaxsinaxlim lim( .a) =a.lim =ax ax ax = +) x 0 x 0 x 0 x 0 x 0 sinaxsinax bx axsinax bx ax alim lim( . . ) lim .lim .limsinbx ax sinbx bx ax sinbx bx b = = = +) x 0 x 0 x 0 x 0tgax sinax a sinax alim lim( . ) lim .lim ax 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 : 13x 01+sinax - cosaxL lim1- sinbx - cosbx=Bi gii : 13x 0 x 01+sinax - cosax 1- cosax+sinax L lim lim1- sinbx - cosbx 1- cosbx - sinbx

= = = 2x 0 x 02ax ax axax ax ax2sin sin cos2sin +2sin cos2 2 22 2 2=lim limbx bx bxbx bx bx2sin- 2sin cos2sin sin- cos2 2 22 2 2 | | |\ .| | |\ .+= =

x 0 x 0ax ax axsin sin cosa2 2 2=lim .limbx bx bxbsin sin- cos2 2 2 += Vy 13aLb= V d 14 : 142x 01 cosaxL limx= Bi gii :www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 12 2 222 2 2142 2x 0 x 0 x 0 x 0ax ax ax2sin sin sin1 cosax a a a2 2 2L lim lim lim . limax axx x 2 2 22 2 (| | | | ( || ( = = = = = || ( || (\ . \ . Vy 214aL2=V d 15 : 15 201 xsinx - cos2xL limsin xx+=Bi gii : 15 2 2001 xsinx - cos2x (1 - cos2x) xsinx L lim limsin x sin xx x = =+ +=

22 2000 0022sin x xsinxsinx(2sinx x) 2sinx xlim lim limsin x sin x sin xx xlim 2 lim 2 1 3sin x sin xx x xx x = = = =| |= + = + = |\ .+ + ++ = Vy L15 = 3 V d 16 : *16 2x01- cosx.cos2x...cosnxL lim(n N )x= eBi gii : 16 2x02x0 1- cosx.cos2x...cosnxL limx1-cosx+cosx-cosxcos2x+...+cosx.cos2x...cos(n-1)x-cosx.cos2x...cosnxlimx===

2x02 2 2x0 x0 x0...1-cosx+cosx(1- cos2x)+...+cosx.cos2x...cos(n-1)x(1- cosnx)limx1-cosx cosx(1-cos2x) cosx.cos2x...cos(n-1)x(1- cosnx)lim lim limx x x == + + + Theo kt qu bi 14 ta c : 22x0121-cosxlimx= 22 2x0 x0 x0.cosx(1-cos2x) 1-cos2x 2lim limcosx lim2 x x = = www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 13 2x022x0 x0 x0 x0. ... .cosx.cos2x...cos(n-1)x(1- cosnx)limx1- cosnx nlimcosx limcos2x limcos(n-1)x lim2 x == = Do 2 2 2 2 2 2161 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 :217 2x01 x cosxL limx+ =Bi gii : 2 217 2 2x0 x0( 1 x cosx 1 x 1) (1 cosx)L lim limx x = =+ + + = 22 2 22 2 2x0 x0 x0 x0 2 2x( (2(2sin1 x 1 1 cosx 1 x 1) 1 x 1)lim lim lim limx x xx 1 x 1) += + = + =++ + ++ 2222x0 x0 x0 x0 2 2 2x x2 2.(2sin sin1 x 1 1 1lim lim lim limx2 xx 1 x 1) 1 x 12 | | |= + = + = |+ + |\ .++ + 1 112 2= + = Vy L17 = 1. Kt lun : kh dng v nh i vi hm s lng gic, hc sinh cn nm vng v vndnglinh hotccphp bin i i s, lng giccng nh p dng cc gii hn c bn. y ch c gii hn x0sinxlim 1x=c s dng trc tip, cc kt qu cn li khi lm bi phi chng minh li. www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 14 vn dng gii hn x0sinxlim 1x= , cn a hm s cn tnh gii hn v dng: 0 0 0xx xx xxsinf (x) f (x) tgf (x)lim , lim ,limf (x) sinf (x) f (x) vi 0xxlim f (x) 0= bngcch thm,bt,ibinhaynhn,chiangthivimtlngthchhpno. Trongkhigiibitp,hcsinhcthgpkhkhn,lngtngavcc dng trn. Gio vin cn khc phc bng cch cho hc sinh lm cc bi tp nh : 2201sinx sin(x 1)lim , lim ,...1 cosx x 3x+2x x Bi tp t luynTnh cc gii hn sau : 1) 01+sinx 1 sinxlimtgxx 2) 0(a+x)sin(a+x) asinalimxx 3) x01 cosxcos2xco3xlim1 cosx4)2202sin x+sinx 1lim2sin x 3sinx+1x 5) 33x41 cotg xlim2 cotgx cotg x 6) 3x01 cosx cos2x cos3xlim1 cos2x 6. Gii hn dng v nh 00 ca hm s m v lgarit. Phng php : Thc hin cc php bin i v s dng cc gii hn c bn sau y : +) xx01lim 1xe =+) x0ln(1 x)lim 1x+=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.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 15 +) x xlnax0 x0a 1 1lim lim .lna lnax x.lnae | | | |\ . = =( V xlnax01lim 1xlna =e) +) x0 x0 x0a.log (1 x)ln(1 x) ln(1 x) 1lim lim lim lnax x.lna lna x = =++ +=V d p dng : V d 18 : ax bx18x0L limxe e=Bi gii : x0 x0ax bx ax bx181) 1)lim lim( (Lx x = = =e e e e ax bxx0 x0ax bxx0 x0( 1) ( 1)lim limx x( 1) ( 1)a. lim b. limax bxa be ee 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 khix 0 thax 0 , do vy ax bxx0 x0( 1) ( 1)lim 1, lim 1ax bx = =e e. V d 19 : sin2x sinx19x0L limsinxe e=Bi gii : sin2x sinx sin2x sinx19x0 x01) 1) ( (L lim limsinx sinxe e e e = = = sin2x sinxx0 x0sin2x sinxx0 x01 1lim limsinx sinx1 1lim .2cosx limsin2x sinxe ee e | | | |\ . = = = = x0 x0 x0sin2x sinx. (2cosx)1 1lim lim limsin2x sinx2 1 1 | | | |\ . = == =e e www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 16 Vy L19 = 1. V d 20 : x 220x 22 xL limx 2= Bi gii : x 2 x 220x 2 x 24) 4) 2 x (2 (xL lim limx 2 x 2 = = =

x 2 x 2x2 x2 x2 x2x 2x2 x21) 2)(x+2) 4 4144(2 (x 2 xlim lim lim limx 2 x 2 x 2 x 22lim lim(x+2) 4ln2 4x 2 = = == = Vy L20 = 4ln2 - 4 V d 21 : 22 3 2x21 2x01 xL limln(1+x )e+ =Bi gii :

2 23 3 2 2x 2 2x21 2 2x0 x0( 1) 1 x 1) ( 1 xL lim limln(1+x ) ln(1+x ) = =+ + =e e

2 23 3 2 2x 2 2x2 2 2x0 x0 x0( 1) 1 1 x 1) ( 1 x 1lim lim limln(1+x ) ln(1+x ) ln(1+x ) = = =+ + e e 3 3 2 233 2 2 23222 22 2x0 x0 22x( ( ) 1( ) 1.1 2xln(1+x )1 x 1)( 1 x 1 x )lim lim( 1 x 1 x )ln(1+x )2x + ++ +| | | |\ .= =+ + ++ +e 2223 2 2 23222xx0 x0 x0 2.( ) 11 2xln(1+x )xlim lim lim2x( 1 x 1 x )ln(1+x ) + +== + +e

222 23 2 23222xx0 x0 x0 x0 2. .( ) 1x 1ln(1+x )2xln(1+x )1 7.1 1.( 2)3 31lim lim lim lim2x1 x 1 x + +== = =+ +eVy 217L3=Kt lun :www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...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 cbn. 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 0f(x) f(x)axx xx xx xxln 1+f(x) log 1+f(x)1 a 1lim , lim , lim , limf(x) f(x) f(x) f(x)e vi 0xxlim f (x) 0=Bi tp t luynTnh cc gii hn sau :1)2) x xx xx054 3lim9 3) x022x3 cosxxlim4) x3 4x0(1 )(1 cosx)lim2x 3xe + 5) x01 1 xlim .lnx 1 x | | | |\ .+6) sin2x sinx2x0lim5x + tg xe e II. GII HN DNG V NH Gii hn dng v nh c dng l : 0xx(x )f(x)L limg(x) =trong : 0 0xx xx(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 1m m 1 1 0n n 1xn n 1 1 0a x +a x +...+a x+aL limb x +b x +...+b x+b =vi *m na ,b 0, m,nN = eKhi xy ra mt trong ba trng hp sau : www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...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:0 m 1 1n n 1m m0 n 1 1 n nn n 1x xmna a aa ax x xb b b b bx x xalim limb+ +...+ +L+ +...+ + = = =+) m > n (bc ca t ln hn bc ca mu, k = m), chia c t v mu cho xm ta c : nm nm n+1nm n0 m 1 1mm 1mx x0 n 1 1mmbxbxa a aaax xxlim limb b bx x x+ +...+ +L+...+ + + = = =+) m < n (bc ca t nh hn bc ca mu, k = n), tng t nh trn ta c : 0 m m 1n m n n m+1x0 n 1n na a a...x x xlim 0b bb ...x xL + + += =+ + + Hc sinh cn vn dng kt qu : 0 0 0 0x x x x x x x x1 1limf (x) lim 0, limf (x) 0 limf (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 vih min{m, n}. 2) Nu f(x), g(x) l cc biu thc c cha cn thc th ta quy c ly gi tr mk(trongklbccacnthc,mlsmcaonhtcaccshng 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 nhvitrnghpf(x),g(x)lccathc.Quahcsinhcthddng phn on kt qu gii hn dng cn tm. V d p dng : V d 22 : 3 222 3x2x 3x 1L lim5x 6 += www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...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 : 333 222 3x x3 122x x655x2x 3x 1L lim lim5x 6 = += += Vy 222L5= . Ta c th trnh by theo cch sau : 3333333 222 3x x x3 13 1x 22x x 2x x65 65x 5xx2x 3x 1L lim lim lim5x 6 | | |\ .| | |\ .= + += = += V d 23 : 2 22x233x (2x 1)(3x x+2)2x+1 4xlim L | | | |\ . + =Bi gii :2 2 4 22 2x x233x (2x 1)(3x x+2) 12x (2x+1)(3x x+2)2x+1 4x 4x (2x+1)lim lim L | |= = | |\ . + + = 3 23 22 3x x5x x+24x5 1 244 1x x x48 28+x4xlim lim8x ++ + += = = =

Vy 2312L =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 5x1 2 3 4 51 1 1 1 1x x x x x 1515xlim | || || || || | | | | | |\ . \ .\ .\ .\ .| | |\ . = = Vy 24 51L5=www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 20 V d 25 : 25x 2x+3L limx 1 =+ Bi gii : Chia c t v mu cho x ta c : 25x x 2 23x+3xL lim limx 1 x 1x1+ = =+ + V phi a x vo trong cn bc hai nn ta xt hai trng hp :*) 2x x > 0 xx + =Khi : x + x + x + 2 22213 3 3x x xlim lim lim 1x 1 x 1xxx1+ 1+ 1+1 = = =+ ++ *) 2x x < 0 xx = Khi , ta c : x x x 2 22213 3 3x x xlim lim lim 1x 1 x 1xxx1+ 1+ 1+1 = = =+ ++ V x x2 21, 1 lim limx+3 x+3x 1 x 1+ = = + + nn khng tn ti x 2x+3limx 1 + V d 26 : 3 2 2265 x 4 4 49x 1 x 4L lim16x 3 x 7 + +=+ + Bi gii : Chia c t v mu cho x ta c : 3 2 23 2 2265 5 x x 4 4 4 4 4 49x 1 x 49x 1 x 4x xL lim lim16x 3 x 7 16x 3 x 7x x = =+ ++ +=+ + + + 233x 4 4559x 1 1 4x x xlim16x 3 1 7x x x =+ + ++ Tng t Bi 25, ta xt hai trng hp : www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 21 *) 4 4 2, x x x > 0 xx x+ = =Khi : 233 2 3+264x x 4 455 4 535 49x 1 1 4 199 0 3x x x x xL lim lim216 03 16x 3 1 7x x x x x1 4x1 716x+ ++=++ += = =+ ++ + ++ *) 4 4 2, x x x < 0 xx x = =Khi ta c : 2 244 42233 333 326x x x 4 4 4 44 55 555 51x x3xx9x 1 1 49x 1 1 4 1 49x xx x x x xL lim lim lim1 716x 3 1 7 16x 3 1 716x xx x x x x = = = =++ + + ++ + + +++ +

243344 55x1x3x1 499 0 3x x21 7 16 016x xlim+ = = =+ ++ V +26 26L L=nn ta c : 263L2=Kt lun :So vi dng v nh 00, 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 ktqu giihncn tm.Ch i vigiihn dng ca hmsccha cn thc ta khng nhn lin hp. y l im khc bit cn phn bit trnh nhm ln. Vigiihnkhix ,cnluhaikhnngx +vx 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 luyn1) ( ) ( )( )( )2 32 2 x2x 3 4x+7lim3x 1 10x 9+ +2) 20 3050x(2x 3) (3x+2)lim(2x+1) www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 22 3) 2 nn+1xn2(x+1)(x 1)...(x 1)lim(nx) 1 ( + ++4) 2x 2x 2x 3xlim4x 4 x+2+ ++ 5) 3 4 5 24 x 4 3x 1 x 2limx 1 x 2+ ++ +6) 33 4xln(1 x x)limln(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 : ( )227xL lim x x x+= + Bi gii : Nhn v chia biu thc lin hp tng ng l : 2x x+x + , ta c : ( )2 2227x 2 x( x x x)( x x+x)L lim x x xx x+xlim+ +=+ += + =+ 2 22 2 x xxx x+x x x+xx x xlim lim+ += =++ + Vx + nn chia c t v mu cho x ta c : 2 x x1 121x x+x1 1xxlim lim+ += =++ + Vy 271L2=Trong v d ny, bng cch nhn lin hp, ta chuyn gii hn cn tm t dng sang dng . www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 23 V d 28 : 28xL lim x+ x x+ ( ( = Bi gii : 28x x( x+ x x)( x+ x x)L lim x+ x x limx+ x x+ + ( = = ( += +

x xx+ x x xlim limx+ x x x+ x x+ += = =+ +

x x1x 1 1lim lim2x+ x x1+ 1x+ += = =++ ( chia c t v mu chox ) Vy 281L2=V d 29 : 229xL lim x x 3 x ( ( = + +Bi gii : Trong v d ny cn lu khix cn xt hai trng hp x+ vx +) Khix+ th : 2 2x x 3 x x x 3 + + + + +Do 2xlim x x 3 x+ ( = ( + + + +) Khix thgii hn c dng. Ta kh bng cch nhn lin hp bnh thng 2 22x x 2( x x 3 x)( x x 3 x)lim x x 3 x limx x 3 x ( = ( + + + + + =+ 2 2x x x 2 2 231x x 3 x x 3lim lim limx x 3 x x x 3 x x x 31xx = = = ++ ++ + + Khix thx < 0, do 2x x = www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 24 x x 223 31 11 1x xlim lim2 21 3x x 31 11x xx = + += =+ + Vy 2xlim x x 3 x+( ( + + = +, 2x1lim x x 3 x2 ( ( + + =Qua v d ny mt ln na nhn mnh cho hc sinh ch vi gii hn khi x cn xtx+ vx i vi hm s cha cn thc bc chn. V d 30 : 3 3 2 230xL 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 3 3 2 2 3 2 230x x) L lim x 3x x 2x lim ( x 3x x ( x 2x x)+ + ((= = (( = + + x x3 2 2 31 2lim lim x 3x x x 2x x G G+ + ((= = (( + +) ( )3 2 32 13 2 3 2 2 3 33 2 3x xx 3xGx 3x x x 3x xxlim x 3x x lim+ ++ (= = = + + + ++

( )2 23 2 3 2 2 3 33 3x x3 3 3133 3 x 3x x x 3x x1 1 1x xxlim lim+ += = = =| |+ + + ++ + + + |\ . +) 2 222x x2x 2x xx 2x xG lim x 2x x lim+ + (= = = + 2 x x2 2 212 2 x 2x x1 1x xlim lim+ + = = = = + + Vy L30 = G1 - G2 = 2 V d 31 : *31m nx 1m nL lim ,(m, nN )1 x 1 x(= e ( Bi gii :www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 25 31 m n m nx 1 x 1m n m 1 n 1L lim lim1 x 1 x 1 x 1 x 1 x 1 x ( ( | | | |= = = || ( ( \ . \ .

1 2m nx 1 x 1m 1 n 1lim lim G G1 x 1 x 1 x 1 x | | | |= = || \ . \ . +) 2 m 11m mx 1 x 1m 1 m (1 x x ... x )G lim lim1 x 1 x 1 x + + + +| |= = = | \ . 2 m 1mx 1(1 x) (1 x ) ... (1 x )lim1 x + + + = = m 2m 1x 1(1 x) 1 (1 x) ... (1 x ... x )lim(1 x)(1 x ... x ) ( + + + + + + + = = + + + m 2m 1x 11 (1 x) ... (1 x ... x ) 1 2 ... m 1 m 1lim1 x ... x m 2+ + + + + + + + + + = = =+ + + Tng t ta tnh c 2n 1G2=Vy 31 1 2m 1 n 1 m nL G G2 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 00. 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 thngchuyn chng v cc dng v nh d tnh hn l 00, . Bi tp t luyn 1) xlim x x x x+ (+ + ( 2) 2 23 3xlim (x 1) (x 1) (+ 3) xlim x x x x x x++ + ( ( 4) 3 2xlim x 1 x (+ www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 26 5) | |xlim ln(5x 8) ln(3x 5)++ + 6)5xlim (x 1)(x 2)...(x 5) x++ + + ( IV. GII HN DNG V NH0. 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) 0, lim g(x) = = kh dng v nh ny, ta thng tm cch chuyn chng v dng gii hn khc d tnh hn nh 00, bng cch nhn lin hp, thm bt, i bin V d p dng :V d 32 : ( )232xL lim x x 5 x+ (= + ( Bi gii : Ta kh dng v nh ny bng cch nhn lin hp a v dng v nh ( )2 22322 x xx( x 5 x)( x 5 x)L lim x x 5 x limx 5 x+ ++ + + (= + = = ( + + www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 27

2 22 2 2 x x xx(x 5 x ) 5x 5lim lim limx 5 x x 5 x x 5 xx+ + ++= = =+ + + + + + 2x x > 0 xx + =Do : 2 x x25 5 5lim lim2 5x 5 x1 1xx+ += =+ ++ + Vy 325L2=V d 33 : 33x 1xL lim(1 x)tg2t= Bi gii : tt 1 x = ta c :x 1 t 0 33t 0 t 0(1 t) tL lim t.tg lim t.tg2 2 2 tt t ( ( | |= = = | ( ( \ . t 0 t 0 t 0tt t 2 22lim t.cotg lim limt t2tg tg2 2 tt (= = = = (t tt t Vy 332L =t Bi tp t luyn 1) ( )2xlim x 4x 9 2x (+ + ( 2) ( )2 4 4xlim x 3x 5 3x 2 (+ ( 3) ( )2 3 3xlim x 4x 5 8x 1 (+ ( 4) x4lim tg2x.tg x4t t (| | | (\ . 5) ( )2 2x axlim a x tg2at ( ( 6) 1 12x xxlim x e e 2 (| |+ (|\ . V. GII HN DNG V NH 1 Dng tng qut ca gii hn ny l :www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 28 | |0g(x)x xlim f (x), trong 0 0x x x xlimf (x) 1, limg(x) = = Hai gii hn c bn thng c s dng khi tnh gii hndng v nh 1 l : +) xx1lim 1xe| | |\ .+ =(1) +) ( )1xxlim 1 x e+ =(2) Trong qu trnh vn dng, hc sinh bin i v dng 0x xf(x)1lim 1f(x)e | | |\ .+ =nu 0x xlimf (x)= ( )01g(x)x xlim 1 g(x) e+ =nu 0x xlimg(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 xlimf (x) a 0= >2) 0x xlimg(x) b=th | |0g(x)bx xlim 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 dngV d 34 : ( )1x34x 0L lim 1+ sin2x= Bi gii :( ) ( ) ( )sin 2x1 1 sin 2x 1x.x sin 2x x sin 2x34x 0 x 0 x 0L lim 1+ sin2x lim 1+ sin2x lim 1+ sin2x (= = = ( www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 29 Ta c : ( )1sin 2xx 0lim 1+ sin2x e=( hc sinh d hiu nn t t = sin2x)

x 0 x 0sin2x sin2xlim 2lim 0x 2x = =Do : ( )sin 2x1x2sin 2x34x 0L lim 1+ sin2x e(= = ( V d 5 : 4 3x35xx 1L limx 2++| |= |+\ . Bi gii : s dng gii hn c bn ta bin i : x 1 11x 2 (x 2)+= ++ + 4 3x(x 2).4 3x(x 2)35x xx 1 1L lim lim 1x 2 (x 2) + ++ +| | +| |= = + ||+ +\ .\ . V(x 2)xx x x1lim 1 e(x 2)434 3x 3x 4xlim lim lim 32(x 2) x 21x +++ + +| |+ = | +\ . = = = + ++nn 335L e =Bi 36 : tg2 y436t 0L lim tg y4t | | |\ . t (| |= | (\ . Bi gii :ty x ,x y 04 4t t= .Ta c :

21 tg ytg2 y2tgy 436t 0 t 01 tgyL lim tg y lim4 1 tgy t | | |\ . | |t ( | |= = = ||(+\ . \ .

222tgy 1 tg y.1 tg y 1 tgy1 tgy 2tgy 2tgy 2tgyt 0 t 02tgy 2tgylim 1 lim 11 tgy 1 tgy ++ (| | | | (= + = + || (+ +\ . \ . ( www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...Nhng dng v nh thng gp trong bi ton tm gii hn ca hm s 30 V1 tgy 2tgyy 02tgylim 1 e1 tgy+| | + = |+\ . v

( )2y 0 y 02tgy 1 tg y. 1 tgy 11 tgy 2tgylim lim | | = + = |+\ . nn 136L 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 li bin v thm bt. Bi tp t luyn 1) ( )2cot g x2x 0lim 1 x+2) 1sin xx 01 tgxlim1 sinx+| | |+\ . 3) 2x22xx 3limx 2| | + |\ .3) ( )cot g xx 1lim 1 sin xt+ t5) 21xx 0lim(cos2x)6) xx1 1lim sin cosx x| |+ |\ . www.TOANTRUNGHOC.com - Ton cho hc sinh v gio vin Trung Hc Ph Thngwww.TOANTRUNGHOC.com - Thi - p n - Chuyn -Bi Tp - Phn Mm Ton,...