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GII HN DY S
http://e-learning.hcmut.edu.vn/
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DY S H!"
Dy s l tp hp cc s c nh ch st nh n ln trong tp hp s t nhin N.
#D: ! "n# n$% n # &% % $% '
$! "n# !n% n # % $% '
(! )"n* l c+p s c,ng- % /0% /$0% '
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1c cch cho 0y s
2 1% !n nx n x n= =
! D2ng li3t 4-
5D- 0y % $% (%'6 0y % !$% !(%'
$! D2ng t7ng 8inh-
)"n* cho 02ng 9i:; thch c? 9in n.
(! D2ng @;y n2p-
A h2ng i s; t>nh thBo cc s h2ng i trc
5D-
5D- 0y 21 11 1% n n nx x x x+= = +
0y 11 2 11 12
% % n nnx x
x x x +
= = =
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D$% &'n &i(u
)"n* l 0y tCng"n"n/% i 8Ei n ? ln)"n* l 0y gi=8"n"n/% i 8Ei n ?
ln
Dy tng v dy gim gi chung l dy n iu.
F 0+; G # G trong Hnh nghI t gEi l tCngJgi=8K ngLt.
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.Mt hi3; s- "n/O "nJso i G&PK
$.Mt thQng s- "n/!"nJso i GPK
J0Rng cho 0y s 0QngK
(.Mt 2o h8 c? h8 s SJ"K% iSJnK # "n
ThQng php 4h=o st 0y Qn i3;-
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#) d*
1 11 2= + + +K! -na x n
1 1/ 1 1 :
2nb x
n
= K
gi=81 111
+ = +
n
n
x
x n
1
1
1
+ =
+n nx x
n
0> tCng
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2 3/ :
3 4n
nc x
n
=
Fi:; th
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D$% +, chn
)"n* l 0y 9H chLn trn U - "n U% n N&
)"n* l 0y 9H chLn 0i 8 - "n 8% n N&
)"n* 9H chLn )"n* 9H chLn trn 9H chLn 0i
VD: Xet tnh b chancua cac day
( ){ }/ 1 nc n
{ }/ 3nb { }2
1/a
n
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{ } { }1 2 3 4 5 6 7 8, , , , , , , , , ,n nx x x x x x x x x x=
L L
DY "N
1ho )"n*% chEn r cc s h2ng t 0y ny
cch tRy X thBoth t0 ch1 23 t4ng d5n t
c 0y con c? )"n*.
#D:
)"$n O *)"$n*
)"$nY* # )"% "(% "Z% '*
)"$n* # )"$% "[% "\% '*
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GiI HN DY S
]Hnh nghI Qn gi=n- )"n* c^ gii h2n l
4hi n r t
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#) d*
lim 11n
n
n =+1
1
1 1
n
nx a
n n
= =
+ +1 11 1
1nx n
n
< < + >+
1h nh ch+t 0y h,i t_
VDy h,i t_ th 9H chn.Vn& nth &
V
n f cth
nfci n
N
&
Y /
cn% n N&
Y /
n% n N& &
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"6c ph7p t86n tr9n d$% hi t*
( )
( )( )
lim , lim
lim lim lim
lim lim lim lim 0
lim lim 0 & lim 0n
K :
K :
n nn n
n n n nn n n
n n n n nn n n n
n n n
n n n
x y
x y x y
x y x y y
x x x x
=
= =
li8 tngJhi3;% t>ch% thQng% cCn%'K #tngJhi3;'K li8
Jh; h2nK
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S HOI TU VA DAY CON
5D- 0y )"n* # )JOKn* phn 4j
2
2 1
1 1
1 1V 2 day conn
n
x
x
= =
li8 "n# UEi 0y con c? {"n}k;
Dy {"n}phn 4j 1 day con phan !y"
2 day con co #$% nhau
2
2 1
nn
n
x ax
x a
a3 @;=-
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GII HAN KEP
1ho ( 0y {"n}% {yn}% {mn}
0
lim lim
n n n
n nn n
x y z n N
x z a
= =
lim
=nn
y a
n n nx y z
a3 @;=- 0 & lim 0 lim 0n n n n
n n
x y n y x
= =
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Dy phn 4j r cRng
ii h2n # - 4hng th: "t "nO q
Dy 4hng h,i t_ gEi l 0y phn 4j-
hng c^ giih2n
Thn 4j r cRng
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0 0lim , : ,&'n nn
x M N N x M n N
= + >
0 0lim , : ,& 'n nn
x M N N x M n N
= <
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#) d*
lim 2n
n = +
22 logn M n M> >
1h >
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"6c ph7p t86n tr9n d$% ph;n <
1lim 0n
na =limn
n a = ! N; th
$! N; th limnn
a
= +JK
lim nn
a
= ,lim nn
b c
=
lim nn
a
= +
lim ( )
lim ,
n nn
n nn
a b
a b
+ =
= neu c 0
, lim nn
b
= + lim ( )n nn
a b
+ = +
lim n nn
a b
= lim nn
a
= + , lim nn
b
= +
>/
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GII HAN C BAN
3 / lim 1,n
nn
=
lim 0, 1n
n
na
a
= >
4 / lim 1, 0nn
a a
= >
1 lim
1 1 lim 0
n
n
n
n
a a
a a
> = < < =
0 lim
0 lim 0n
n
n
n
> =
< =
2/.Ha!"#
$/.%!"&'()a#
lim 0, 0!
n
n
aa
n= >
ln5 / lim 0, 0
p
nn
n
= >
ln p nn n a( (
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2/ lim 1
=nn
n
2/ lim
= n
a n 1 21/ lim lim 0n n
b nn
= =
/ lim 2
= +nn
c 1/ lim 02
=
n
nd
2
/ lim 03
=nn
nf
#) d*
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? DNG #@ ABNH
0, 0 , ,
0
0 01 ,0 ,
V]i i [ php ton c,ng% tr% nhn% chi-
V]i i 02ng 8W ( ) nb
na
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2
sin2 / lim 1n
n n
n +
1000
3 / lim
n
n n
!1/ lim nn
n
n ! 1 2 10 0nn nn n n nn < = KK
2 2
sin
0 01 1
n n n
n n + +
#Ci n EEE: 1000 10 0
2
n n
n
<
#) d* tFng hp
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eng c+p s nhn
( )1 1
2 1 1lim 1 lim
1 1
n !n
n n
!! ! !
! !
+
