Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 1 Giới Hạn Cơ Bản Hàm Số 1. íℎ ℎấ ơ ả ủ ớ ℎạ : 0 0 0 0 0 0 0 0 0 lim ( ) , lim ( ) a) lim ( ) , R; b) lim ( ) ( ) , lim ( ) ( ) c) lim ( ).g(x) ( ) d) lim , ( 0) ( ) lim ( ) > 0 e) li lim ( ) x x x x x x x x x x x x x x x x x x f x a g x b f x a f x g x a b f x g x a b f x ab f x a a g x b f x a g x b 0 0 0 ( ) m ( ) ( ) ( ) ( ) ) lim ( ) lim ( ) , lim ( ) g x b x x x x x x x f x a f x g x h x f g x a f x a h x a 2. ộ ố ớ ℎạ ơ ả: 0 0 0 0 0 1 0 1 1 1 0 0 sin )lim 1 0 sin tan )lim 1 0 tan 1 c)lim 1 0 1 1 1 d)lim 0 1 1 ln(1 ) e)lim 1 0 ln 1 f)lim 1 1 )lim 1 lim 1 x x x x x x x x x x x x x x a x x x x x b x x x x e x e x x x x x x x x x x x x x e g x e x e e Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 2 1 h) lim 1 )limsin khong ton tai x x x e x g x 3.Các dạng vô định thường gặp : 0 0 0 ) ; b) ; c)0. d) e)1 )0 ) 0 a f g Dạng: 0 0 Có dạng: 0 ( ) lim ( ) x x f x g x trong đó ( ) ( ) f x g x = 0 0 , Cách làm: Ta tìm cách triệt tiêu đi 0 x x Ví dụ1 : 2 3 2 2 2 2 2 2 4 8 lim lim lim 2 4 12 2 2 x x x x x x x x x x x Ví dụ 2 : 1 1 lim 1 x x x 1 1 1 1 1 1 1 lim lim 1 1 1 x x x x x x x x x Ví dụ 3: 1 1 lim 1 x x x 1 1 1 1 1 1 1 lim lim 1 1 1 x x x x x x x x x Ví dụ 4 : 1 1, 1 1 lim 1 1, 1 x x x x x Ví dụ 5: 4 2 lim 5 x x x x x Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 3 4 2 2 2 2 4 2 2 thi 2 2 lim lim 2 5 x x x x x x x x x x x x x x x x Ví dụ 6: 3 3 2 2 lim 3 2 x x x x x Cách 1 : 3 3 3 2 2 3 2 2 2 2 3 3 2 2 2 2 2 33 2 3 2 2 3 2 2 3 3 2 2 lim 3 2 lim 3 2 3 3 lim 3 lim lim 1 3 3 2 2 lim 2 lim lim 1 2 2 suy ra : lim 3 2 1 1 2 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Cách 2 : 1 3 1 2 ln ln 1 3 3 2 2 3 2 3 1 1 3 2 lim 3 2 lim 1 1 lim lim 1 1 1 1 2 x x x x x x x x x x x x x x xe xe x x xe xe Ví dụ 7: 2 0 cos cos lim sin m n x ax bx cx Cách 1: 2 2 2 22 2 2 2 2 2 2 0 0 0 2 22 2 2 2 2 2 2 2 2 2 2 2 2 0 cos cos cos cos 1 sin 1 sin lim lim lim sin sin sin 1 sin 1 1 sin 1 1 1 sin sin 1 sin sin 2 2 lim sin 2 m n m n m n x x x m n x ax bx ax bx ax bx cx cx cx ax bx ax bx a b a b ax bx m n cx c c m n Cách 2: Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 4 1 1 1 1 ln cos ln cos ln 1 cos 1 ln 1 cos 1 2 2 2 0 0 0 2 2 cos 1 cos 1 2 2 2 2 2 0 cos cos lim lim lim sin sin sin 1 1 1 2 2 lim sin 2 ax bx ax bx m n m n m n x x x ax bx m n x ax bx e e e e cx cx cx a b e e b a m n cx c c n m Ví dụ 8: 2 0 ln cos lim ln 1 x x x 2 2 2 0 0 0 2 2 2 0 0 2 2 2 0 ln cos ln 1 cos 1 cos 1 lim lim lim ln 1 ln 1 ta co': lim 1 cos lim 2sin 2 2 4 2 cos 1 1 2 lim 2 x x x x x x x x x x x x x x x x x x x x Ví dụ 9: 10 9 1 2 1 lim 2 1 x x x x x 9 8 7 6 5 4 3 2 10 9 8 7 6 5 4 3 2 1 1 9 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 1 1 2 1 lim lim 2 1 1 1 1 8 lim 1 7 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Ví dụ 10: lim n n x a x a x a 1 2 3 2 2 1 1 2 3 2 2 1 1 lim lim lim n n n n n n n x a x a n n n n n n x a x a x x a x a xa a x a x a x a x x a x a xa a na Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 5 Ví dụ 11: 2 2 1 1 2 3 1 2 3 2 2 1 1 1 2 3 2 1 2 2 3 1 1 1 1 lim lim 1 1 1 ( 1 1 ( 1) 1 lim lim 1 1 1 1 1 1 1 lim 1 1 1 lim n n x x a n n n n n n x x n n n x n n n x x n x x nx n x x x x x x x n x x x x x n x x x x x x x x x x x x x 3 4 4 5 2 2 3 3 4 4 5 1 1 1 1 1 lim 1 1 1 1 ( 1) 1 2 3 1 2 n n n n n n n n n x x x x x x x x x x x n n n n n Ví dụ 12: 1 2 lim n n n x a x a na x a x a Ta có : 1 1 2 1 1 1 2 1 1 2 2 3 2 3 4 3 2 1 2 2 2 a 1 lim lim 1 2 1 2 n n n n n n n n n n n n n n n n n n n n n n n x a x a x a na x a x a x x a a na x a x a x x a a na x a x x a a a x x a a x a na x a n n a n n a x a Ví dụ 13: 5 0 243 3 lim x x x Cách 1: Ta có tính chất : 0 1 1 lim x u u Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 6 1 5 5 5 0 0 0 3 1 1 3 1 3 243 243 3 3 1 1 243 lim lim lim . 243 5 405 243. 243 x x x x x x x x x Cách 2: Ta chú rằng : lnu e u Suy ra: 1 1 ln 1 ln 1 5 243 5 243 5 5 0 0 243 3 3 3 3 1 3 5.243 405 243 3 1 lim lim 405. 405 x x x x x x x e e x x x x Cách 3 : Lopital 1 405 kq Ví dụ 14: 3 0 1 1 lim x x x x Cách 1: 3 3 0 0 0 2 3 3 3 0 0 2 3 3 0 0 2 3 3 1 1 1 1 1 1 lim lim lim 1 1 1 1 1 1 1 1 1 lim lim 1 1 1 1 1 1 1 1 lim lim 2 3 6 1 1 1 1 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Cách 2: 32 1 1 ln 1 ln 1 3 3 2 0 0 0 1 1 1 1 1 2 3 lim lim lim 6 x x x x x x x x x e e x x e e x x x x Ví dụ 15 : 2 2 1 3 2 4 2 lim 3 2 x x x x x x Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 7 2 2 2 2 1 1 2 2 2 1 1 2 2 2 3 2 4 2 3 2 4 2 3 2 4 2 lim lim 3 2 3 2 3 2 4 2 5 6 1 5 11 6 1 lim lim 2 3 2 3 2 4 2 1 2 3 2 4 2 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Cách 2 : 2 2 4 3 2 1 ln 1 4 3 2 2 2 2 2 1 1 1 2 2 2 2 2 2 1 1 1 1 3 3 1 3 2 4 2 3 2 lim lim lim 3 2 3 2 3 2 3 4 3 4 3 3 6 6 4 3 4 7 3 1 4 2 lim lim lim lim 3 2 2 2 2 2 3 2 2 3 2 x x x x x x x x x x x x e x x x x e x x x x x x x x x x x x x x x x x x x x x x Các dang bài tập tương tự: 3 2 5 7 2 0 0 3 3 5 44 0 0 3 sin 2 3 cos7 cos9 1)lim 2)lim 3)lim 3 1 cos6 ln 1 3 sin ln 1 sin 1 cos . cos5 cos7 1 4)lim 5)lim 1 cos9 8 2 x x x x x x x x x x x x x x x x x xe e x x x x x x 1 2 0 3 4 0 0 2 2 7 sin 0 0 0 1 6)lim sin 1 2 cos cos .cos5 .cos 7 .cos9x 1 7)lim 6) lim tan ln 1 3 3 ln 1 3 5 1 2 cos 1 5 1 sin cos 8)lim 9)lim 10)lim ln(1 ) 1 x x x x x x x x x x e x x x x x x x x x x x x x x x x x x x x x x e 2 2 1 0 6 3 52 34 6 7 3 2 2 03 tan 2 sin sin ln 6 11)lim 12) lim 13) lim 1 2ln sin 3 2 3cos 1 ln 2 2 1 tan 2 14)lim 15)lim 1 sin 2 x x x x x x x x x x x x x x x x x x x 0 5 33 4 2 2 2 5 0 0 1 1 16)lim cos 1 sin 1 cos7 2 1 3sin 3 1 4sin 17)lim 18)lim 19)lim tan 2 sin 2 5 x x x x x x x x x e x x x e e x x x x x x x x Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 8 20)lim , 0 m n x a x a x a x Dạng 2: 0 1 , , 0. Đối với dạng 1 ta có công thức sau : 1 0 lim 1 x x x e hoặc 1 lim 1 x x e x Ví dụ 16: 3 3 1 lim 1 x x x x Cách 1: 3 3 1 3 2 1 2 3 3 2 0 1 3 3 3 3 1 1 2 2 2 lim lim lim 1 lim 1 1 1 1 1 1 x x x x x x x x x x x x x x e e x x x x Cách 2: 3 3 2 2 3 3 ln 1 1 1 3 3 3 1 2 2 1 lim lim lim 1 1 lim 1 lim 1 1 1 1 x x x x x x x x x x x x x x e e x x x Ví dụ 17 : cot 0 lim ln 1 x x x e e Cách 1: cot 1 cot cot ln 1 ln 1 0 0 0 cos cos 1 ln 1 . sin sin 0 0 lim ln 1 lim 1 ln 1 lim 1 ln 1 lim lim x x x x x e e x x x x x x x e x e x e x x x x x e e e e e e e Cách 2: cot cos cos 1 cot ln 1 ln 1 ln 1 sin sin 0 0 0 0 lim ln 1 lim lim lim x x x x x x x e x e x e e x x x x x e e e e e e Ví dụ 18: 1 1 lim x x x e x Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 9 Cách 1: 2 . 1 1 2 1 2 1 1 2 2 lim lim 1 1 lim 1 lim 1 x x x x x x x x x x e e e x x x x Cách 2: 1 1 1 1 2 ln ln 1 1 1 2 ln 1 2 1 lim lim lim lim x x x x e x e x x x x x x x x x x x e e e e e e x Ví dụ 19 : sin 0 lim cos x x x Cách 1: 2 3 2 sin 2sin 2sin sin 2 2 2sin 0 sin 0 0 0 0 lim cos lim cos lim 1 sin lim 1 x x x x x x x x x x x x x e e Cách 2: 2 sin sin ln 1 cos 1 sin ln cos 0 2 0 0 0 0 lim cos lim lim lim 1 x x x x x x x x x x x x e e e e Ví dụ 20 : 1 0 lim , 0 x x x I x a a Cách 1: 1 1 1 lna 1 ln 1 ln 1 0 0 0 lim lim 1 1 lim 1 ln 1 x x a x x x a x x x I x a x a a x e Cách 2: 1 1 1 (ln 1) 1 ln ln 1 1 ln 1 ln ln 1 0 0 0 0 0 lim lim lim lim lim x x x a x a x a x x a x a x x x x x x x x x x I x a e e e e e Bài tập tương tự : 1 sin 1 1 cos 1 cos cos 2 sin2 cos 0 0 0 0 sin 1 3tan 3 1)lim 2)lim 3)lim 1 4)lim 1 tan 1 5sin 2 x x x x x x x x x x x x x x x x x . Nguyễn Duy Đăng Facebook: https://www.facebook.com/ngsaolanhgia Trang 1 Giới Hạn Cơ Bản Hàm Số 1. íℎ ℎấ ơ ả ủ ớ ℎạ : 0 0 0 0 0 0 0 0 0 lim ( ) , lim (