BÀI TẬP GIỚI HẠN DÃY SỐ VÀ HÀMSỐ 1.
Trang 1BÀI TẬP GIỚI HẠN DÃY SỐ VÀ HÀM
SỐ
1 xn= cos(√1
n) + n
2 xn= n
n − 1
3 xn= (2n − 1)!!
(2n)!! .
1 2n + 1
4 xn =
1 −1 2
1 − 1 3
1 − 1 n
5 (x0 = 3
xn+1 = xn
2 + 1
1
x1 = 3
2
xn+1= x2n− 2xn+ 2
2 xn=
r
2 +
q
2 +p2 + +√
2
3 xn= 1 + 1
22 + 1
32 + + 1
n2
1 lim
n→∞
9n
n!
2 lim
n→∞
1 + √1
2 +
1
√
3 + +
1
√ n
3 lim
n→∞
1 − 1
2+
1
4+ + (−1)
n 1
2n
4 lim
n→∞
n + 1
n + (−2)n
5 lim
n→∞
(−1)n−1
ln n − n
6 lim
n→∞
√
n3+ 2n − 1 − 5np3 n3 + 3n + 5
n + 1
7 lim
n→∞
√
n2+ 1 −√3
n3+ 1
8 lim
n→∞
2n+ 3n
2n− 3n
9 lim
n→∞
n cos(nπ) + (n2+ 1) arctan n + n3− 2
2n3+ 1
10 lim
n→∞
2n3+ 3n2− ln9n
3 ln7n − n3
11 lim
n→∞
n
√
3nn2− 2n
1 lim
n→∞nα−2 √
n2− 1 − 2n = 0
2 lim
nto∞nα2+1 √
n2+ 1 − n = +∞
Trang 23 lim
n→∞n2α+3
3
√
n3 + 1 − n2 3n − 2 6= 0
1 lim
x→1
xx− 1
ln x
2 lim
x→ π
2
(1 + cos x)sin2 x1
3 lim
x→±∞xe1x − 1
4 lim
x→0 ±xe1x − 1
5 lim
x→1
m
√
x − 1
n
√
x − 1, với m, n là các số tự nhiên.
6 lim
x→1
√
x +√
x − 1 − 1
√
x2− 1
7 lim
x→π4
√
2 − 2 cos x
π − 4x
8 lim
x→atanπx
2a sin
x − a 2
9 lim
x→∞
5x− 3x
5x+ 4x
10 lim
x→0(cos)x21
11 lim
x→∞
ln (x2+ 2x + 3)
ln (3x4+ 12x + 1)
12 lim
x→0
logax(1 + x)
x
13 lim
x→∞
x − 3
x + 2
2x+1
14 lim
x→0(sin x + cos 2x)1x
15 lim
x→+∞ sin√
x + 1 − sin√
x
16 lim
x→+∞(2 + x)1x
17 lim
x→π4±0
tan(4x − π) 2x −π2
18 lim
x→0
ln cos(−2x)
ln cos 3x
19 lim
x→0 +(e1x + 1
x)x
20 lim
x→2
2x− x2
x − 2
21 lim
x→0
tan(2x) − 3 arcsin(4x) sin(5x) − 6 arctan(7x)
22 lim
x→+∞(x + 2x)1x
23 lim
x→0 +
ex+ ln(1 + 2 sin x) − 1
3
√
8 − x4 − 2
24 lim
x→1
ex−1x − 1
x − 1
25 lim
x→0 ±
ex−1x − 1
x − 1
26 lim
x→0
1 + x cos x −√
1 + 2x ln(1 + x)
27 lim
x→0
etan x− ex
tan x − x
28 lim
x→∞
2x2+ 3 2x2− 1
x 2
1 α(x) = (etan x− 1) sinh(x2− 3x4)
2 α(x) = ln(arcsin x + 1)(√3
x2+ 1 − 1)
3 α(x) = √3
cos 2x −√
cos x
4 α(x) = ex− cosh x
5 α(x) = ex− sin 2x − cosh x
6 αx =√
1 + 2x2 −√3
1 + x2
7 α(x) = (x + 1) tan x − sin x
8 α(x) = (x2+ 1) tan x − sin x
Trang 34.2 So sánh bậc các VCB
1
(
α(x) = esin x− etan x
β(x) = ln(1 + x sin x) , x → 0
2
(
α(x) = cos x − cosh x
β(x) = x2− 2 arcsin x , x → 0
3
α(x) = arctan x
x2
β(x) = e−x
, x → +∞
4
(
α(x) = ex(x+1)−√1 + 2x
β(x) = x2− 2 sinh x , x → 0
5 (α(x) = x arctan x + (x + 1) ln(1 − 2x)
β(x) = 3
q
x2+px3+√4
x12+ x8
, x → 0+
1
(
α(x) = x ln x
β(x) = ln2x x → ∞
2
(
α(x) = x3− 2 ln x
β(x) = 3(x2+ 1) ln x x → ∞
3
α(x) = 1
x β(x) = ln x
x → 0+
4
(
α(x) =q3
x5+px12+ x4√
x β(x) = sin2x3+ 2x
x → ∞
1 lim
x→+∞
ln(1 + x + ex)
x + ex
2 lim
x→2
3
√
x − 1 − 1
lnx 2
3 lim
x→0
(cos x)tan x− 1
x3− 3x4
4 lim
x→0
sin x − cos 2x − 1
cos x − sin 2x − 1
5 lim
x→0
"
(1 + 4x)1x
e4
#1x
6 lim
x→±∞x2(e−x23 − ex21 )
7 lim
x→0
1
x cot πx
2
8 lim
x→±∞(x2− 2)
1 − cos1
x
9 lim
x→+∞x [ln(x + a) − ln x]
10 lim
x→0
3x− 2x
x2 − 2x