PHƯƠNG PHÁP KHỬ DẠNG VÔ ĐỊNH - TOÁN THPT

PHƯƠNG PHÁP KHỬ DẠNG VÔ ĐỊNH - TOÁN THPT

WWW.MATHVN.COM Gi i h n d ng gi i h ng tr c ti ng gi i h ih ih nd ph ng v nh c bi , ta nh c a gi i h n ng g , , , , i dung t ng d ng c th I GI I H N D NH 0 i h n c d u v i bi u th Gi i h n d nh i v t t ph ih nb gi i h n b t nh ng gi i h ng g p nh t i h n d kh ih nh ih nd h c sinh k nh 0 ch n cho n d ng 0 ih nc Nh n d ng gi i h nh gi c xem n 0 cd cd nk , h c sinh c nh N u gi i h nh gi i h n p B i v y vi c nh n d ng cho h m c ph i iv id nh , vi c nh n d c sinh ng g p gi i h n : f(x) x g(x) lim x www.MATHVN.com lim f(x) = lim g(x) = x x0 x x0 WWW.MATHVN.COM f(x) g(x) lim ng h p x x Th c t h c sinh hay g m t s f(x0 ) = g(x0 ) = c sinh ph i th c hi chuy n v 0 gi i h n b ng d nh Khi gi ng d h c sinh vi c nh n d ts : f(x) x g(x) lim x nh n m nh cho lim f(x) x ho c lim g(x) x x x0 ng h 0 nd i ng : u chung c a nh : L1 = lim x ih x-2 x +1 i: L1 = lim x lim : L2 = x x-2 2-2 = x +1 22 x+2 x2 - i: L2 = lim x : L3 = lim x lim(x+2) = 1+2 = x lim(x - 1) = 12 - = x+2 = x2 - 1 x x x i: L = lim x x x = lim x www.MATHVN.com (x-1)(x 2) (x 1)(x+1) lim x (x-2) 1-2 (x+1) 1+1 lim x x 3x +2 x2 1 2 WWW.MATHVN.COM D 0 nh uv i c th sau : f(x) g(x) 0) Lo i : lim x x Kh d v = g(x0) = nh b t x0) t Gi s : f(x) = (x x0).f1 (x - x )f1 (x) f (x) f(x) lim lim lim x x0 g(x) x x0 (x - x )g (x) x x0 g (x) 1 f1 (x) v n g1 (x) N u gi i h n lim x x d x0).g1 nh 0 pl n nh ng : : L4 = lim x 2x - 5x +2 x +x - i: t v chung : x - 2x - 5x +2 (x - 2)(2x - 1) lim x +x - x (x - 2)(x + 3) 2x - 2.2 = lim x x+3 L4 = lim x V y L4 5 : L5 = lim x x - 3x +2 x - 4x + i: x - 3x +2 (x - 2)(x - 1) lim - 4x + x x 2x (x - 2)2 x-1 = lim x 2x-2 L5 = lim i h n c a t b ng 1, gi i h n c a m u b ng 0) V y L4 www.MATHVN.com WWW.MATHVN.COM : L6 lim x x+x +x3 + +x n - n (m, n x+x +x3 + +x m - m i : Ta s N* ) v chung : x 1b x + x2 + x3 + + xn n = (x 1) + (x2 1) + (x3 - 1) + + (xn - 1) x + x2 + x3 + + xm m = (x L6 x+x +x3 + +x n - n x+x +x3 + +x m - m lim x 1) + (x2 1) + (x3 - 1) + + (xm - 1) (x- 1)+(x - 1)+(x3 - 1)+ +(x n - 1) (x- 1)+(x - 1)+(x - 1)+ +(x m - 1) lim x (x- 1) + (x + 1) + + (x n-1+ x n-2 + + x +1) lim x (x- 1) + (x + 1) + + (x m-1+ x m-2 + + x +1) + (x + 1) + + (x n-1+ x n-2 + + x +1) 1 + (x + 1) + + (x m-1 + x m-2 + + x +1) lim x + (1 +1) + + (1n-1+ 1n-2 + + +1) + (1 +1) + + (1m-1 + 1m-2 + + +1) n(n + 1) m(m + 1) 2 n m V y L6 : L7 n(n + 1) m(m + 1) n(n + 1) m(m + 1) 2x - 5x3 +3x + x - 1 3x - 8x + 6x - lim x i: (x-1)(2x - 3x +1) 2x - 5x +3x + x - = lim L7 = lim x 3x - 8x + 6x - x (x-1)(3x - 5x +x+1) 2x - 3x +1 (x-1)(2x - x -1) = lim = lim x 3x - 5x + x +1 x (x-1)(3x - 2x -1) 2x - x -1 (x -1)(2x+1) = lim = lim x 3x - 2x -1 x (x -1)(3x+1) 2x+1 2.1+1 = lim = = x 3x+1 3.1+1 www.MATHVN.com WWW.MATHVN.COM V y L7 = K t lu n: gi x - x0 v h p lo iv ih Ph i n m v ng th cb f(x) = ax + bx + c = (x - x ) ax - t : n b ng th - b = (a - b)(an -1+ an - c b c ba c , ( f(x0) = 0) x0 , h c sinh c n nh n-2 + bn - 1), n N* n an + bn = (a + b)(an -1- an - 2b - abn - 2+ bn - ng th c t h c sinh d nh , c n l ng h p c th n n-1 n-2 ng h c bi t : x - = (x - 1)(x + x Tu ph mt im kh d ng i h n b ng ta v n g p gi i h n d i gi i h nh u) T gi i h n c nh m pt n 0 p t luy n x 3x 1) lim x x 4x (1 x)(1 2x)(1 3x) x 2) lim x x100 2x 3) lim 50 x x 2x 4) lim xn x f(x) g(x) Lo i : xlim x u v i bi u th c (g i t tr x - x0 kh i h n b ng Bi u th c ch www.MATHVN.com (n 1) n (x 1) 0)=g(x0)= t bi u th c ch th p) ng c a u p hay c, nh m kh , m u hay c WWW.MATHVN.COM t uc hay nhi u l cc ih kh d nh n ( A ± B)( A ( A ± B)( A h ng th pm t cs d B) = A - B , (A 0, B 0) A B+ B2 ) =A ± B n cho h c sinh th h c sinh d nh : hai (a - b)(a + b) = a - b2 (a ± b)(a ab + b2 ) = a ± b3 ng: 3x - - x x2 - lim : L8 = x i: t u v i bi u th ng, ta c: L8 = lim x 3x - - x x2 - ( 3x - - x)( 3x - + x) (x - 4)( 3x - + x) lim x 3x - - x lim x (x - 4)( 3x - + x) x+1 lim x (x + 2)( 3x - + x) V y L8 = www.MATHVN.com (x - 2)(-x + 1) x (x - 2)(x + 2)( 3x - + x) 2+1 16 (2 + 2)( 3.2-2+2) lim 16 WWW.MATHVN.COM 9: L9 lim x x+2 x+5 i: L9 lim x ( x+2 1)( x+2 1) ( x+5 2) x+2 lim x+5 = xlim1 = lim x x ( x+5 2)( x+5 2) ( x+2 1) (x + - 1)( x+5 2) (x + - 4)( x+2 1) x+5 x+2 lim1 x (x + 1)( x+5 2) (x + 1)( x+2 1) 2 V y L9 = n 10 : L10 lim m x x -1 , (m, n N* ) x -1 i: n L10 lim m x x -1 x -1 ( n x - 1) ( n x ) n-1 +( n x ) n-2 + + n x +1 ( m x ) m-1 +( m x ) m-2 + + m x +1 = lim ( x - 1) ( m x ) m-1 +( m x ) m-2 + + m x +1 ( n x ) n-1 +( n x ) n-2 + + n x +1 m x (x - 1)(m x m-1 +m x m-2 + +m x +1) (x - 1)( n x n-1 + n x n-2 + + n x +1) = lim x m = lim x V y x m-1 +m x m-2 + +m x +1 m n n-1 n n-2 x + x + + n x +1 n L10 = m n K t lu n: u th d ih hi u.Vi www.MATHVN.com y u gi i h n c nh bi u th ch cs t c, WWW.MATHVN.COM iv ih u th nk nh ih uy i bi u th c c ng k nh i bi n s cho l i gi i ng n g gi N p t luy n 1) lim x3 x x x 2) lim x 4) lim a x 6) lim x x f(x) g(x) iv m u(x) n v(x) f(x) = lim ,(m u(x ) x g(x) x x0 g(x) L= lim x Ta bi n a 0)=g(x0)= S d ng thu p Ch ng h bi u th x2 x x2 n ax 5) lim x x x0 x x n x 3x 2 x b a b 3) lim 2 x a x a Lo i 3: lim x2 n h v(x ) = 0,g(x ) = 0) i: m u(x) - c + c - n v(x) u(x)- n v(x) L lim lim x x0 x x0 g(x) g(x) m u(x) - c n v(x) - c = lim lim x x0 x x0 g(x) g(x) m T i h n L1 b lim x x0 m u(x) g(x) -c n , L2 lim x x0 v(x) - c g(x) c p ng : 11 : L11 www.MATHVN.com lim x x+3 x+7 x 3x+2 WWW.MATHVN.COM i i: x+3 x+7 lim x x 3x+2 L11 ( x+3 2) + (2 x+7) lim x x 3x+2 x+3 2 lim = lim x x 3x+2 x x x+7 3x+2 (2 ( x+3 2)( x+3+2) = lim lim x (x 3x+2)( x+3+2) x (x = lim x x+3 lim 3x+2)( x+3+2) x (x (x x+7) x+7 ( x+7) 3x+2) x+7 ( x+7)2 (x+7) 3x+2) x+7 ( x+7) x 1 x lim (x 1)(x 2)( x+3+2) x (x 1)(x 2) x+7 ( x+7)2 = lim x 1 lim x 2)( x+3+2) (x 2) x+7 ( x+7)2 = lim x = = (x 1 (1 2)( 1+3+2) (1 2) 1+7 ( 1+7)2 1 12 V y L11 6 12 : L12 1+2x - 1+3x lim x x2 i: L12 1+2x - 1+3x lim x x2 =lim x 1+2x - (x+1) + (x+1) - 1+3x lim x x2 1+2x - (x+1) (x+1) - 1+3x +lim x x2 x2 www.MATHVN.com WWW.MATHVN.COM 1+2x - (x+1) = lim x x2 1+2x +(x+1) 1+2x +(x+1) (x+1) - 1+3x (x+1)2 ( x 1) 1+3x ( 1+3x )2 +lim x lim x lim x x (x+1)2 ( x 1) 1+3x ( 1+3x )2 (1+2x) - (x+1)2 x2 lim x 1+2x +(x+1) (x+1)3 - (1+3x) x (x+1)2 (x 1) 1+3x -1 x+3 lim 1+2x +(x+1) x (x+1)2 (x 1) 1+3x -1 1+2.0 +(0+1) 1 2 (0+1) ( 1+3x )2 ( 1+3x )2 0+3 (0 1) 1+3.0 ( 1+3.0) V y L12 K t lu n : i h n c a bi u th c ch c u gi i h n r i t m t h ng s ng ch 0) td mt ng hi u qu iv tm p C ho c v(x0)) hay m t bi u th c Vi ph i th t tinh t Thu d p t luy n 1) lim x 3) lim bx x x x 20 x Gi i h n d www.MATHVN.com 2) lim x m ax 5) lim x x x n x x nh x 11 8x 43 2x 3x 2x x 4) lim x sin x 6) lim x 4x 6x x2 c 10 WWW.MATHVN.COM V y L19 = 2x x lim x x 20 : L20 i: L20 2x x 2 x lim x (2x 4) (x 4) x lim x 4(2x 1) (x 2)(x+2) 2x x2 lim lim lim x x x x x x x x x 2 lim (x+2) 4ln 4 lim x x x lim V y L20 = 4ln2 - x e 2x lim x ln(1+x ) 21 : L21 i: L21 x e 2x lim x ln(1+x ) lim x lim x ( x 1) (e ln(1+x ) 1) ln(1+x ) 2x 2x ( x 1) (e 1) x2 e 2x lim lim x ln(1+x ) x ln(1+x ) x 1)( (1 x )2 x 1) e 2x 2x lim lim x 2x ln(1+x ) ( (1 x )2 x 1)ln(1+x ) x ( lim x ( (1 lim x 03 (1 x2 x2 x )2 1)ln(1+x ) x2 x2 lim x ln(1+x ) 1 x )2 lim x e 2x lim x 2x2 e 2x 2x 2x lim x ln(1+x ) 2x ln(1+x ) lim x 1.( 2) 3 V y L21 K t lu n : www.MATHVN.com 16 WWW.MATHVN.COM ih nd nh c hi c sinh th c u h c sinh ph i ih lu th s d ih n, b h c sinh ph i bi c ih nv m ng : ln 1+f(x) loga 1+f(x) a f(x) ef(x) v i lim f (x) lim , lim , lim , lim x x0 x x0 f(x) x x f(x) x x0 x x0 f(x) f(x) p t luy n i h n sau : 2) lim 1) x 3) lim x 5) lim x 3x 3x 1 x ln x x (1 ex )(1 cosx) 2x3 3x lim 6) x x2 esin2x esinx 5x + tg x NH Gi i h n d L 4x lim 4) x cosx II GI I H N D 9x 5x nh lim x (x f(x) x0 g(x) ) lim f(x) x x (x ) lim g(x) x x (x ) kh d t cho lu th a b c cao nh t c a t uc c u f(x) C th g(x) 1) N h k f(x), g(x) cho x v i k = max{m, n} L lim x a m x m +a m 1x m + +a1x+a v i a m ,bn bn x n +bn 1x n + +b1x+b0 y m www.MATHVN.com 0, m,n N* ng h p sau : 17 WWW.MATHVN.COM L c: x u b ng nhau), chia c t a a a a m + m + + n1 + x x n lim a m x lim x x b b b bn bn + n + + n1 + n x x x u cho xn ta +) m = n (b c c a t +) m > n (b c c a t l c: m L am bn c c a m u, k = m), chia c t a0 am a + + m1 + m x x x lim x bn b1 b0 bn + m n + m n+1 + + x x x xm am + +) m < n (b c c a t nh lim x u cho am bn xm n cc am : a am n m+1 xn x b bn x xn am n m L xlim x bn H c sinh c n v n d ng k t qu : lim f (x) lim 0, lim f (x) x x x x f (x) x x 0 ng h cc at h nc v i lim x x0 f (x) c sinh c n t t qu gi i u cho xh chia t h min{m, n} 2) N tr u th m k cc cl tc cc cao nh ch h ng cc at (m u th ng h im u t qu gi i h n d ng c c d ng : 22 : L22 www.MATHVN.com lim x 2x3 3x 5x3 18 WWW.MATHVN.COM u cho x3 i : Chia c t L22 3 x lim x x3 x3 V y L22 L22 3x 5x 2x lim x c: lim x 3x 5x 2x 3 23 : L23 lim x x x3 lim x x3 x3 x3 x lim x x3 x3 5 3x (2x 1)(3x x+2) 2x+1 4x i: L23 lim x 3x (2x 1)(3x x+2) 2x+1 4x lim x 4x 5x x+2 8x3 4x x x2 8+ x lim x lim x 12x (2x+1)(3x x+2) 4x (2x+1) x3 2 V y L23 24 : L24 lim x (x 1)(x 2)(x 3)(x 4)(x 5) (5x 1)5 i: L24 lim x (x 1)(x 2)(x 3)(x 4)(x 5) (5x 1)5 lim x 1 1 x x x x x V y L24 x 55 55 www.MATHVN.com 19 WWW.MATHVN.COM 25 : L25 x+3 x2 lim x i: Chia c t c: 1+ x+3 lim x x2 L25 x lim x x x ng h p : *) x x>0 x2 x 1+ x lim x 1+ + x x lim x + x x *) x x x0 9x x lim x 16x x L+ 26 *) x L26 lim x x x3 x x5 x x 1 x x 51 16 x x4 L+ 26 L26 x4 x2 lim x 16 x4 x2 , x x x3 x5 x3 x5 x0 lim 5x lim 5 V y L32 33 : L33 x i: x L33 tt x t lim t.tg t lim t.cotg t x lim(1 x)tg (1 t) t lim t.tg t t lim t tg t t 2 t lim t t tg 2 V y L33 p t luy n 4x 2x 1) lim x x 4x 3) lim x x 8x a x x tg 2a V GI I H N D D ng t www.MATHVN.com 3x x 4) lim tg2x.tg x 5) lim a x 2) lim x 6) lim x x e x x e 3x x NH a gi i h 27 WWW.MATHVN.COM lim f (x) x g(x) lim f (x) 1, lim g(x) x x0 Hai gi i h x0 cs d ih n d e (2) n d ng, h c sinh bi lim e n u x x f (x) 0 lim g(x) i v d ng f(x) lim x x f(x) g(x) lim g(x) e n u x0 bi cc nh e (1) x +) lim x x x0 x +) lim x x x x x x0 i gi i h n c c sinh v n d ng m f(x), g(x) tho sau (d u ki n : 1) lim f (x) a x x0 2) lim g(x) b x x0 g(x) lim f (x) x x0 ab Hai gi i h ih nd nh ng 34 : L34 lim 1+ sin2x x x i: L34 lim 1+ sin2x x www.MATHVN.com x lim 1+ sin2x x sin 2x sin 2x x lim 1+ sin2x x sin 2x sin 2x x 28 WWW.MATHVN.COM sin 2x lim 1+ sin2x x sin 2x x lim x L34 x sin 2x x sin 2x e2 3x x lim x x i: s d ng gi i h n ta bi i: (x 2) L35 3x x x lim x lim x 3x (x 2) lim x lim x lim tg t e 3x x tg2 lim tg t lim t www.MATHVN.com 4 e3 y ,x y y 2tgy tgy L35 x lim x x y t y x i: 3x (x 2) (x 2) tg2 L36 (x 2) (x 2) lim 1 (x 2) x L36 t t = sin2x) 0 : L35 x 1 x sin 2x 2x 2lim lim 1+ sin2x x h c sinh d hi e lim t tgy tgy tg y 2tgy lim t y tg y 2tgy 2tgy tgy tgy 2tgy 2tgy tg y tgy 2tgy 29 WWW.MATHVN.COM tgy 2tgy 2tgy lim y tgy 2tgy tg y tgy 2tgy lim y e L36 e lim tgy y 1 K t lu n : V id nh , vi c nh n d p, h c sinh ph i v n d ng t m t hai gi i h i bi t ih nc ch y cs d i v i h c sinh p t luy n 1) lim x x 2 cot g x x www.MATHVN.com sin x x2 x2 3) lim x x 5) lim(cos 2x) tgx 2) lim x sin x 3) lim sin x cot g x x 1 x2 6) lim sin x x cos x x 30 ... + 1) 2x - 5x3 +3x + x - 1 3x - 8x + 6x - lim x i: (x-1)(2x - 3x +1) 2x - 5x +3x + x - = lim L7 = lim x 3x - 8x + 6x - x (x-1)(3x - 5x +x+1) 2x - 3x +1 (x-1)(2x - x -1 ) = lim = lim x 3x - 5x +... x 3x - - x x2 - ( 3x - - x)( 3x - + x) (x - 4)( 3x - + x) lim x 3x - - x lim x (x - 4)( 3x - + x) x+1 lim x (x + 2)( 3x - + x) V y L8 = www.MATHVN.com (x - 2)(-x + 1) x (x - 2)(x + 2)( 3x - +... + (x3 - 1) + + (xm - 1) (x- 1)+(x - 1)+(x3 - 1)+ +(x n - 1) (x- 1)+(x - 1)+(x - 1)+ +(x m - 1) lim x (x- 1) + (x + 1) + + (x n-1+ x n-2 + + x +1) lim x (x- 1) + (x + 1) + + (x m-1+ x m-2 + +