Chuyên đề Giải tích Lớp 11 - Chương 4 - Bài 5: Giới hạn khi x tiến tới vô cực (Có lời giải)

 GIỚI HẠN KHI x TIẾN TỚI VÔ CỰC
Câu 1: Tìm các giới hạn sau:
 3x2 x 7 (4x2 1)(7x 1) x x 3
a). lim b). lim c). lim 
 x 2x3 1 x (2x3 1)(x 3) x x2 x 2
 x 2x3 x
d). lim (x 1) e). lim x .
 4 2 5 2
 x 2x x 1 x x x 3
 LỜI GIẢI
 2 1 7 1 7
 x 3 3 
 3x2 x 7 x x2 2 3
a). lim lim lim x x lim 0 
 x 3 x x x 
 2x 1 3 1 1 2x
 x 2 x 2 
 x3 x3 
 2 1 1 
 x 4 x 7 
 (4x2 1)(7x 1) x2 x 
b). lim lim 
 x 3 x 
 (2x 1)(x 3) 3 1 3 
 x 2 x 1 
 x3 x 
 1 1 
 4 7 
 x2 x 28
 lim lim 0
 x 1 3 x 2x
 x 2 1 
 x3 x 
 3 3
 x x 1 1 
 x x 3 x x 1
c). lim lim lim x x lim 0 . 
 x 2 x x x 
 x x 2 2 1 2 1 2 x
 x 1 x 1 
 x x2 x x2 
 x x
d). lim (x 1) lim (x 1) 
 x 2x4 x2 1 x 1 1 
 4 
 x 2 2 4 
 x x 
 1 1 1 1 
 lim (x 1) lim (x 1) lim x 1 0
 3 
 x 2x x x x 2 x x x x 2 
 3 1 
 x x 
 2x3 x x2 2
e). lim x lim x lim x 
 x 5 2 x x 2
 x x 3 5 1 3 x
 x 1 
 x3 x5 
 2 2
 lim x lim x 2
 x x x x Câu 2: Tìm các giới hạn sau:
 3x2 x 3 x4 x3 11
a). lim 2x2 1 x b). lim c). lim 
 x x x 4 x 2x 7
 2x4 x2 1 x4 x 2x4 7x3 15
d). lim e). lim f). lim .
 x 1 2x x 1 2x x x4 1
 LỜI GIẢI
 2 2 1 2
a). lim 2x 1 x lim x 2 x lim 2x x
 x x 2 x 
 x 
 lim x 2 x lim 2x x lim x 2 1 
 x x x 
 2 1 3 
 x 3 
 3x2 x 3 x x2 
b). lim lim lim 3x 
 x x 4 x 1 x 
 x 1 
 x 
 1 11 
 4 
 4 3 x 1 3 
 x x 11 x x 1 3
c). lim lim lim x 
 x 2x 7 x 7 x 2
 x 2 
 x 
 4 1 1 1 1
 x 2 x 2 
 2x4 x2 1 x2 x4 2 4 2x
d). lim lim lim x x lim 
 x 1 2x x 1 x 1 x 2
 x 2 2
 x x
 4 1 1
 x 1 x 1 
 x4 x x3 3
e). lim lim lim x 
 x 1 2x x 1 x 1
 x 2 2
 x x
 4 7 15 
 x 2 
 2x4 7x3 15 x x3 
f). lim lim 2 
 x 4 x 
 x 1 4 1 
 x 1 
 x4 
Câu 3: Tìm các giới hạn sau:
 2 2 2 x 3 2 x 3
a). (x 1) (5x 2) b). c). 
 lim 4 lim lim
 x (3x 1) x x2 x 5 x x2 x 5
 2
 2x5 x3 1 x2 x 2x x x x
d). lim e). lim f). lim
 x (2x2 1)(x3 x) x 2x 3 x x 10
 LỜI GIẢI 2 2 2 2
 2 1 2 2 1 2 
 x 1 x 5 1 5 
 (x 1)2 (5x 2)2 x x x x 25
a). lim lim lim 
 x 4 x 4 x 4 81
 (3x 1) 4 1 1 
 x 3 3 
 x x 
 6 2 2 2 
 x 1 x3 1 1 
 x6 2 x6 3 x3 1
b). lim lim lim x lim 
 x 3 x x x 1
 3x 1 3 1 3 1 3
 x 3 x 3 3 3
 x3 x3 x
 3 3
 x 2 2 
 2 x 3 2x 3 x 
c). lim lim lim lim x 2
 x 2 x x 1 5 x 1 5
 x x 5 2 1 5 1 
 x 1 x 1 2 2
 x x2 x x x x
 5 1 1 1 1
 x 2 2 
 2x5 x3 1 x2 x5 2 3
d). lim lim lim x x 1
 x 2 3 x x 
 (2x 1)(x x) 2 1 3 1 1 1 
 x 2 x 1 2 1 
 x2 x2 x2 x2 
 1 1 
 x2 1 2x x 1 2 1
 2 1 2
 x x 2x x x 
e). lim lim lim lim x 1
 x 2x 3 x 3 x 3 x 3
 x 2 x 2 2 
 x x x
 1 1 
 2 x 1 1 1
 2 x x 1 1 1 
 x x x x x 
f). lim lim lim lim x 1
 x x 10 x 10 x 10 x 10
 x 1 x 1 1 
 x x x
Câu 4: Tìm các giới hạn sau:
 2x3 x x 1 x2 2x 3 4x 1
a). lim x b). lim (x 2) c). lim
 5 2 3 
 x x x 3 x x x x 4x2 1 x 2
d). lim x2 4x x e). lim 3 8x3 1 2x 1
 x x 
 LỜI GIẢI
 3 1 1
 x 2 2 
 2x3 x x2 2
a). lim x lim x lim x x 
 x x5 x2 3 x 1 3 x 
 5 2 1 3
 x 1 3 5 x 1 
 x x x3 x5 1
 2 2
 lim x 2
 x 1 3
 1 
 x3 x5
 1 1 
 x 1 1 
 x 1 2 x 2 
b). lim (x 2) lim x 1 lim x 1 x 
 x 3 x x 
 x x x 3 1 x 1 
 x 1 
 2 x 1 2 
 x x 
 1 
 1 
 2 x
 lim 1 1.
 x x 1 
 1 
 x2 
c).
 2 2 3 2 3 
 x 1 4x 1 x 1 4x 1
 x2 2x 3 4x 1 x x2 x x2 
 lim lim lim
 x 2 x x 1
 4x 1 x 2 2 1 
 x 4 x 2 x 4 2 x 2
 x2 x
 2 3 1 1
 x 1 4x 1 x 1 4 3 
 x 2 x 4x 1 x 3
 lim x lim lim lim x 
 x 1 x 4x x 2 x 2 x 2 5
 x 4 x 2 x 4 1 5 
 x2 x x
 x2 4x x 4x
d). lim x2 4x x lim lim 
 x x 2 x 
 x 4x x 2 1 
 x 1 x
 x 
 4x 4x
 lim lim 2
 x x x x 2x
 8x3 1 8x3
e). lim 3 8x3 1 2x 1 lim 1 
 x x 2
 3 8x3 1 3 8x3 1.2x 4x2
 1
 lim 2 1
 x 
 3 1 3 1 2
 3 x 8 3 x 8 .2x 4x
 3 3 
 x x 
 1 1 1
 lim 1 lim 1 lim 1 1
 x 2 x 2 2 2 x 2
 3 3 3 3 2 4x 4x 4x 12x
 4 x 2 x .2x 4x
Câu 5: Tìm các giới hạn sau: a). lim x2 4x x b). lim x2 1 x c). lim x2 3x 2 x
 x x x 
d). lim 4x2 4x 1 2x 3 e). lim x2 4x 1 2x 3
 x x 
 LỜI GIẢI
 x2 4x x2 4x
a). lim x2 4x x lim lim 
 x x 2 x 
 x 4x x 2 4 
 x 1 x
 x 
 4x 4x
 lim lim 2 .
 x x x x 2x
 x2 1 x2 1
b). lim x2 1 x lim lim 
 x x 2 x 
 x 1 x 2 1 
 x 1 x
 x2 
 1 1
 lim lim 0 .
 x x x x 2x
 x2 3x 2 x2 3x 2
c). lim x2 3x 2 x lim lim 
 x x 2 x 
 x 3x 2 x 2 3 2 
 x 1 x
 x x2 
 2 
 x 3 
 x 3x 3
 lim lim .
 x x x x 2x 2
 4x2 4x 1 4x2 
d). lim 4x2 4x 1 2x 3 lim 3 
 x x 4x2 4x 1 2x 
 1 
 x 4 
 4x 1 x 4x
 lim 3 lim 3 lim 3 2 3 5 .
 x x x 2x
 2 4 1 2 x 2x
 x 4 2x
 x x 
 x2 4x 1 4x2
e). lim x2 4x 1 2x 3 lim 3 
 x x x2 4x 1 2x
 2 4 1 
 x 3 
 3x2 4x 1 x x2 8x2
 lim 3 lim 3 lim 3 
 x x x x 2x
 2 4 1 x 2x
 x 1 2x
 x x2 
 8x2
 lim 3 lim( 3x) 3 .
 x 3x x 
Câu 6: Tìm các giới hạn sau: a). lim x2 4x 3 x2 3x 2 b). lim 4x2 9x 21 4x2 7x 13 
 x x 
 x3 x2 
c). lim 
 2 
 x 3x 4 3x 2 
 LỜI GIẢI
 x2 4x 3 x2 3x 2
a). lim x2 4x 3 x2 3x 2 lim 
 x x x2 4x 3 x2 3x 2
 x 1 x 1
 lim lim
 x 4 3 4 3 x 1 3 4 3 
 x2 1 x2 1 x2 1 1 
 2 2 2 2 
 x x x x x x x x 
 1 1 
 x 1 x 1 
 x x 1
 lim lim 
 x 1 3 4 3 x 1 3 4 3 2
 x 1 1 x 1 1 
 2 2 2 2 
 x x x x x x x x 
 4x2 9x 21 4x2 7x 13
b). lim 4x2 9x 21 4x2 7x 13 lim 
 x x 4x2 9x 21 4x2 7x 13
 34 
 x 2 
 2x 34 x 
 lim lim
 x 9 21 7 13 x 9 21 7 13 
 x2 4 x2 4 x2 4 4 
 2 2 2 2 
 x x x x x x x x 
 34 34 
 x 2 x 2 
 x x 1
 lim lim 
 x 9 21 7 13 x 9 21 7 13 2
 x 4 4 x 4 4 
 2 2 2 2 
 x x x x x x x x 
 x3 x2 
c). L lim 
 2 
 x 3x 4 3x 2 
 3 2 2
 x3 x2 x 3x 2 x 3x 4 2x3 4x2
 lim lim lim
 x 3x2 4 3x 2 x 3x2 4 3x 2 x 3x2 4 3x 2 
 2x3 4x2 
 x3 4
 3 2 
 x 
 lim lim x . 
 x 3x2 4 3x 2 x 4 2 
 x2 x 3 3 
 2 2 x
 x x x 
 4 4 2 2 2
Do lim lim lim 0 nên L .
 x x x x2 x x2 3.3 9