TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 10 tháng 06 năm 2010 BTVN NGÀY 10-06 Tính các giới hạn sau đây. ( ) 3 2 x 2 x 0 2 x 0 2 x 0 x sin x sin x 2 x 0 x x 4 x 1 Bµi1: lim x cos 4 sin sin sinx 2 Bµi 2 : lim x 1 cosx cos2x 3 Bµi 3: lim x 1 cosx cos 2x cos 2010x 4 Bµi 4 : lim x ln sin x cosx 5 Bµi 5 : lim x e cos2x 6 Bµi 6 : lim x x 3 7 Bµi 7 : lim x 1 8 Bµi8 → → → → →∞ − → →+∞ − − π − − − − − + − − − +   −  ÷ +   − ( ) 3 3 2 2 x 3 x 2 2 x 0 3 x 0 3 2 x : lim x 3x x x 1 tan x sin x 9 Bµi 9 : lim x 1 x cos x 10 Bµi10 : lim x 1 tan x 1 sin x 11 Bµi11: lim x x x 2 12 Bµi12 : lim sin(x 1) →+∞ →∞ → → →∞ + − − + − − + − − + − + − + − − − ………………….Hết………………… BT Viên môn Toán hocmai.vn Trịnh Hào Quang Hocmai.vn – Ngôi trường chung của học trò Việt 1 TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 28 tháng 02 năm 2010 HDG CÁC BTVN • BTVN NGÀY 09-06: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x 0 x 0 x 0 x 0 x 0 m n x 1 m 1 m 2 x 1 1 x 1 2x 1 3x 1 *Bµi1: lim x 1 x 1 2x 1 3x 1 x 1 2x 1 x 1 2x 1 x 1 x 1 lim x 1 x 1 2x 1 3x 1 1 x 1 2x 1 x lim x 3x 1 x 1 2x 2x 1 x x 3 1 x 1 2x 2 1 x 1 lim lim 1 2 3 6 x 1 x 1 *Bµi 2 : lim x 1 x 1 x x lim → → → → → → − − → + + + − + + + − + + + + + − + + + − = + + + − + + + − + = + + + + + + + + + + = = = + + = − − − + + = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) m 1 m 2 n 1 n 2 n 1 n 2 x 1 100 50 x 1 100 99 98 50 49 48 x 1 x 1 x 1 x x x 1 m lim n x 1 x x x 1 x x x 1 x 2x 1 *Bµi3 : lim x 2x 1 x 1 2(x 1) x 1 x x x 1 2 98 49 lim lim 48 24 x 1 2(x 1) x 1 x x x 1 2 − − − − − − → → → → + + + + + + = = − + + + + + + + + − + − + − − − − + + + + − = = = = − − − − + + + + − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 20 2 10 x 2 3 20 20 20 10 20 20 20 10 10 20 10 x 2 x 2 x 2 2 2 x 0 x 0 x 0 x x 2 *Bµi 4 : lim x 12x 16 x 2 (x 1) x 2 (x 1) x 2 (x 1) 3 lim lim lim (x 2) (x 4) 2 (x 2) (x 4) (x 2) (x 4) x 9 x 16 7 *Bµi 5: lim x x 9 3 x 16 4 x lim lim lim x x x 9 3 → → → → → → → − − − + − + − + − +   = = = =  ÷ − +   − + − + + + + − + − + + − = = + + + ( ) ( ) ( ) x 0 x 0 x 0 x x x 16 4 1 1 1 1 7 lim lim 6 8 24 x 9 3 x 16 4 → → → + + = + = + = + + + + Page 2 of 8 TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 28 tháng 02 năm 2010 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 x 0 3 3 x 0 x 0 x 0 x 0 x 0 2 3 3 3 2 x 0 3 2 2 x 0 x 0 2 1 x 8 x *Bµi 6 : lim x 2 1 x 1 8 x 2 2 1 x 1 8 x 2 lim lim lim x x x 2x x 1 13 lim lim 1 12 12 x 1 x 1 x 8 x 2 8 x 4 2x 1 1 3x *Bµi 7 : lim x 2x 1 (x 1) 1 3x (x 1) 2x 1 (x 1) lim lim x x → → → → → → → → → + − − + − − − − + − − − = = − − = − = + =   + + − + − +  ÷   + − + + − + − + − + + − + = = ( ) ( ) ( ) ( ) ( ) 2 3 2 2 x 0 x 0 2 2 2 3 3 2 x 0 2 2 2 3 3 3 4 5 x 0 3 x 0 2x 1 (x 1) 1 3x (x 1) x lim lim x 2x 1 (x 1) x 1 3x (x 1) 1 3x (x 1) x (x 3) 1 3 lim 1 2 2 x 1 3x (x 1) 1 3x (x 1) 1 4x. 1 6x. 1 8x. 1 10x 1 *Bµi8 : lim x 1 4x. 1 6x. 1 8x lim → → → → → + + + + − + − − =   + + + + + + + + +  ÷   − + − = − − = −   + + + + + +  ÷   = + + + + − + + + ( ) ( ) ( ) 4 5 3 4 3 4 3 3 x 0 3 4 5 3 4 x 0 x 0 3 x 0 x 0 2 x 0 . 1 10x 1 4x. 1 6x. 1 8x x 1 4x. 1 6x. 1 8x 1 4x. 1 6x. 1 4x. 1 6x 1 4x 1 4x 1 lim x 1 4x. 1 6x. 1 8x. 1 10x 1 1 4x. 1 6x. 1 8x 1 lim lim x x 1 4x 1 6x 1 1 4x 1 lim lim x x 1 4x 1 XÐt : I lim li x → → → → → → + − + + + + + + − + + + + + − + + + − + + + + + − + + + − = + + + − + − + + + − = = ( ) x 0 n n 5 4 3 2 x 0 4x 4 m 2 2 x 1 4x 1 1 2nx 1 Còng nh$ vËy ta cã : I lim 2 I I I I I 8 x → → = = + + + − = = ⇒ = + + + = Page 3 of 8 TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 28 tháng 02 năm 2010 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 x 0 3 3 2 2 x 0 x 0 x 0 x 0 x 0 2 2 2 2 3 3 3 4 x 7 4 4 4 t 2 2x 1 x 1 *Bµi 9 : lim sin x 2x 1 1 x 1 1 x 1 1 2x 1 1 lim lim lim sin x sin x sin x 2 x lim lim 2 0 2 sin x sin x 2x 1 1 x 1 x 1 1 x x x 2 x 20 *Bµi10 : lim x 9 2 t § Æt t x 9 x t 9 I lim → → → → → → → → + − + + − − + − + − + − = = − = − = − =   + + + + + +  ÷   + − + + − = + ⇒ = − ⇒ = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 4 3 4 4 4 t 2 t 2 t 2 4 2 4 t 2 t 2 4 2 3 4 4 3 2 t 2 2 3 4 4 3 3 x 0 7 t 11 t 2 t 7 3 t 11 3 t 16 lim lim lim t 2 t 2 t 2 t 7 3 t 4 t 2 t 16 lim lim t 7 3 t 2 t 11 3 t 11 9 t 4 t 2 16 32 176 lim 3 27 27 t 11 3 t 11 9 1 4x 1 6x *Bµi11: lim → → → → → → → − − + − − − + − − = − = − − − − + + + − − =   − + − + + + +  ÷   + + − = + =   + + + +  ÷   + − + ( ) ( ) ( ) ( ) 2 2 3 2 2 2 x 0 x 0 x 0 3 2 2 x 0 x 0 2 2 2 3 3 2 x 0 2 2 2 3 3 x 1 4x (1 2x) 1 6x (1 2x) 1 4x (1 2x) lim lim lim x x x 1 4x (1 2x) 1 6x (1 2x) x lim lim x 1 4x (1 2x) x 1 6x (1 2x) 1 6x (1 2x) 4x (3 2x) lim x 1 6x (1 2x) 1 6x (1 2x) → → → → → → + − + + − + + − + = − = + + + + − + − − =   + + + + + + + + +  ÷   − + −   + + + + + +   1 12 7 2 3 2 = − + = ÷  • BTVN NGÀY 10-06: Page 4 of 8 TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 28 tháng 02 năm 2010 ( ) 2 x 2 t 0 t 0 t 0 t 0 t 0 x 0 x 0 4 x 1 Bµi1: lim x cos 4 t(t 4) t(t 4) § Æt : t x 2 x t 2 I lim lim t t 1 sin cos 4 4 2 t(t 4) t(t 4) (t 4) 16 lim lim lim t t sin . . t 4 4 4 . t 4 4 sin sinsinx 2 Bµi 2 : lim x sin sin sinx lim sin si → → → → → → → → − − π + + = − ⇒ = + ⇒ = = − π π   +  ÷   + + + = − = − = − = − π π π π π π − = ( ) ( ) ( ) ( ) 2 x 0 2 2 2 x 0 x 0 x 0 2 2 2 2 x 0 x 0 x 0 sin sinx sinx . . 1 nx sin x x 1 cosx cos2x 3 Bµi 3 : lim x cosx 1 cos 2x 1 cos x cosx cosx cos2x 1 cosx lim lim lim x x x x 2 sin cosx 1 cos 2x 1 2 cosx.sin x 2 lim lim lim 2 x x 1 cos 2x 1 cos2x x 4. 2 → → → → → → →   =  ÷   − − − − + − − = = + − = + = + + +    ÷   2 1 3 1 2 2 = + = ( ) 2 x 0 2 x 0 2010 2 2 x 0 x 0 2 2 n 1 2 2010 2 2 2 1 cos xcos2x cos 2010x 4 Bµi 4 : lim x 1 cos x cosx cos x cos2x cos x cos 2x cos 2010x lim x cosx. 1 cos2x 1 cos x lim lim I x x nx 2 sin 1 cos nx n 1 2 XÐt I I I I I x 2 4 nx . n 2 → → → → − − − + − + + = − − = + + + − = = = ⇒ = + + + =    ÷   ( ) 2 2 2 1 2 3 2010 2 + + + + Page 5 of 8 TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 28 tháng 02 năm 2010 ( ) ( ) ( ) ( ) ( ) x 2 x 0 x 0 x 0 t 0 x 0 2010(2010 1)(2.2010 1) 12 ln sin x cosx 5 Bµi 5 : lim x ln sin x cosx ln sin x cosx sin 2x lim lim . 2x sin 2x 2x ln sin x cosx ln 1 t sin 2x Mµ : lim lim 1 Víi t sin 2x vµ lim 1 sin 2x t 2x I 1.1 1 →∞ → → → → → + + = + − + + = = + + = = = = ⇒ = = cosx cos3x 2 x 0 cosx cos3x 2 2 x 0 cosx cos3x cos x cos3x 2 2 x 0 x 0 cosx cos3x 2 2 x 0 e cos 2x 6 Bµi 6 : lim x e 1 1 cos2x lim x x e 1 e 1 cos x cos3x *)Ta c ã : lim lim . x cosx cos3x x e 1 1 cos3x 1 cosx lim cos x cos3x x x − → − → − − → → − → − −   − − = +  ÷     − − − =  ÷ −   − − −   = −  −   cosx cos3x t x 0 t 0 2 2 2 2 x 0 2 x 0 x x x x e 1 e 1 . Do lim lim 1 cos x cos3x t 1 cos3x 1 cos x 3 1 lim 4 x x 2 2 1 cos2x *)MÆt kh¸c : lim 2 I 4 2 6 x x 3 7 Bµi 7 : lim x 1 2 2 1 lim 1 . §Æt : x 2t 1;x t x 1 x 1 t − → → → → →+∞ →+∞ − − = = ÷ − − −   − = − =  ÷   − = ⇒ = + = +   −  ÷ +     = + = ⇒ = − → +∞ ⇒ → +∞  ÷ + +   ⇒ 2t 1 2t 1 2 t t t 1 1 1 I lim 1 lim 1 . lim 1 e t t t − − →+∞ →+∞ →+∞       = + = + + =  ÷  ÷  ÷       Page 6 of 8 TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 28 tháng 02 năm 2010 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 2 2 x 3 3 2 2 x 2 3 3 2 2 x x 3 3 2 3 2 2 3 2 x 3 3 2 x x x 2 2 8 Bµi 8 : lim x 3x x x 1 lim x 3x x x x 1 x A B 3x *)A lim x 3x x lim x 3x x x 3x x 3 lim 1 3 3 1 1 1 x x 1 1 x 1 x B lim x x 1 x lim lim 1 1 x x 1 x 1 1 x x →+∞ →+∞ →+∞ →+∞ →+∞ →+∞ →+∞ →+∞ − + − − + = + − − − + − = − = + − = + + + + = =   + + + +  ÷   − + − + = − + − = = − + + − + + 3 x 0 2 2 3 2 x 0 x 0 x 0 2 2 x 0 2 2 2 2 x 0 x 0 1 2 tan x sin x 9 Bµi 9 : lim x 1 sinx x sinx 1 (1 cos x) 2sin 1 cosx x 2 lim lim lim x x .cosx 2 x 4. .cos x 2 1 x cosx 10 Bµi10 : lim x 1 x 1 cosx 1 1 lim lim x x 1 x 1 → → → → → → → = −    ÷  ÷   − −   − −  ÷   = = = =    ÷   + − −     + − − = − =  ÷  ÷  ÷ + +     2 2 x 0 x 2sin 1 1 2 lim 1 2 2 x 4. 2 →    ÷ −  ÷ − = + =  ÷    ÷  ÷     Page 7 of 8 TRUNG TÂM HOCMAI.ONLINE P.2512 – 34T – Hoàng Đạo Thúy Tel: (094)-2222-408 Hà Nội, ngày 28 tháng 02 năm 2010 ( ) ( ) ( ) 3 x 0 3 3 x 0 x 0 2 2 x 0 3 2 x 1 3 2 x 1 1 tan x 1 sin x 11 Bµi11: lim x tan x sin x s inx(1 cos x) lim lim x 1 tan x 1 sin x x 1 tan x 1 sin x cosx x sin sinx 2 .2 x x 4. 1 1 1 2 lim . 2 2 4 1 tan x 1 sin x cosx x x 2 12 Bµi12 : lim sin(x 1) x 1 x lim → → → → → → + − + − − − = = + + + + + +    ÷   = = = + + + + − − − − + = ( ) 2 2 x 1 x 1 x 1 (x 1)(x x 1) x 1 (x 1) 1 (x x 1) (x 1) lim lim sin(x 1) sin(x 1) sin(x 1) x 1 sin(x 1) Do lim 1 I 5 x 1 → → → − + + + − + − + + + + = = − − − − − = ⇒ = − ………………….Hết………………… BT Viên môn Toán hocmai.vn Trịnh Hào Quang Page 8 of 8 . (094)-2222-408 Hà Nội, ngày 10 tháng 06 năm 2010 BTVN NGÀY 10-06 Tính các giới hạn sau đây. ( ) 3 2 x 2 x 0 2 x 0 2 x 0 x sin x sin x 2 x 0 x x 4 x 1