0 Ths.NGUYN QUC TIN
NGUYN QUC TIN
BI GING TON CAO CP
THNH PH H CH MINH-2011
1
1 Ths.NGUYN QUC TIN
CHNG 1. GII HN-TNH LIN TC CA HM S
1.1 Gii hn hm s
1.1.1 nh ngha Cho hm s ( )f x xc nh trong mt ln cn ca 0x (c th tr ti 0x ). S L c gi l
gii hn ca hm s ( )f x khi x dn n 0x nu:
00, 0, : (0 ( ) )x D x x f x L
v c k hiu
0
lim ( )x x
f x L
hay ( )f x L khi 0x x .
Gii hn ca hm s ( )f x khi x dn n 0x cn c th nh ngha thng qua gii hn ca dy
s nh sau:
0
0lim ( ) ( ):n n nx x f x x x f xL x L
1.1.2 nh l Cho ( ), ( ), ( )f x u x v x xc nh trong mt ln cn ca 0x c th tr ti 0x .
Nu ( ) ( ) ( )u x f x v x vi mi x thuc ln cn v 0 0
lim ( ) lim ( )x x x x
u x v x L
th
0
lim ( )x x
f x L
V d. Chng minh 0
sinlim 1x
x
x
Tht vy :02
x x ta c bt ng thc sincos 1xxx
, m 0
lim cos 1x
x
suy ra
0
sinlim 1x
x
x
1.1.3 Mt s tnh cht ca gii hn hm s
i) Nu 0
lim ( )x x
f x L
th gii hn l duy nht
ii)0
limx x
C C
(C : hng s)
iii) Nu ( ) ( ),f x g x x thuc mt ln cn no ca 0x hoc v cc th
0 0lim ( ) lim ( )x x x x
f x g x
(nu cc gii hn ny tn ti).
2 Ths.NGUYN QUC TIN
iv) Nu ( ) ( ) ( ),f x g x h x x thuc mt ln cn no ca 0x hoc v cc v
0 0lim ( ) lim ( )x x x x
f x L h x
th 0
lim ( )x x
g x L
v) Gi s cc hm s ( ), ( )f x g x c gii hn khi 0x x khi ta c cc kt qu sau :
0 0 0
lim ( ( ) ( )) lim ( ) lim ( )x x x x x x
f x g x f x g x
lim ( ) lim ( )x x x xo o
kf x k f x
lim ( ). ( ) lim ( ). lim ( )x x x x x xo o o
f x g x f x g x
00 0
0
,
lim ( )( )lim lim ( ) 0( ) lim ( )
x x
x x x xx x
f xf x g xg x g x
1.2 V cng b Gi s ta xt cc hm trong cng mt qu trnh, chng hn khi ox x . (Nhng kt qu t
c vn ng trong mt qu trnh khc)
1.2.1 nh ngha Hm ( )x c gi l mt v cng b (VCB) trong qu trnh ox x nu
0lim ( ) 0
x xx
V d. sin , , 1 cos x tgx x l nhng VCB khi 0x , cn 212
xx
l VCB khi x
1.2.2 So snh hai VCB Cho ( )x v ( )x l hai VCB trong mt qu trnh no (chng hn khi ox x ). Khi
tc tin v 0 ca chng i khi c ngha quan trng. C th ta c cc nh ngha:
Nu ( )lim 0( )xx
th ta ni ( )x l VCB bc cao hn VCB ( )x trong qu trnh ( ( )x
dn ti 0 nhanh hn ( )x khi ox x )
Nu ( )lim 0( )x Lx
th ta ni ( )x v ( )x l hai VCB ngang cp trong qu trnh ( ( )x
v ( )x dn ti 0 ngang nhau khi ox x .
c bit khi 1L ta ni ( )x v ( )x l hai VCB tng ng, k hiu l ( ) ( )x x .
1.2.3 Mt s VCB tng ng c bn khi 0x
sin x x tgx x arcsin x x ; arctgx x
3 Ths.NGUYN QUC TIN
2( )1 cos2
axax 1log (1 )ln
x xa a 1 1x x ln(1 ) x x
-1 lnxa x a -1xe x 11 ... , ( , 0)n n p p
n n p p pa x a x a x a x n p a
Sinh vin c th t kim tra cc tng ng ny (xem nh bi tp)
V d. So snh cp ca cc VCB:
( ) sin ; ( ) 1 cosx x tgx x x , khi 0x
Ta c:
0 0 0 0
1sin 1( ) sin sincoslim lim lim lim 0( ) 1 cos 1 cos cosx x x x
xx x tgx xxx x x x
Do , ( )x l VCB cp cao hn ( )x
V d. So snh cp ca cc VCB: 2( ) 1 cos , ( ) , 0x x x x x
Ta c: 20 0
( ) 1 cos 1lim lim 0
( ) 2x xx x
x x
Do , ( )x v ( )x l hai VCB cng cp.
1.2.4 Quy tc ngt b VCB cp cao i) Nu 1( ) ( )x x v 1( ) ( )x x trong cng mt qu trnh th trong qu trnh y
1
1
( )( )lim lim
( ) ( )xx
x x
ii) Cho ( )x v ( )x l hai VCB trong mt qu trnh v ( )x c cp cao hn ( )x . Khi
( ) ( ) ( )x x x .
T hai kt qu trn ta suy ra quy tc ngt b VCB cp cao:
Gi s ( )x v ( )x l hai VCB trong mt qu trnh no . ( )x v ( )x u l tng ca
nhiu VCB . Khi gii hn ca t s ( )( )xx
bng gii hn ca t s hai VCB cp thp nht
trong ( )x v ( )x .
V d. Tm cc gii hn sau:
1) 2 3
3 80
3sin 4sinlim
5xx x x
x x x
Ta c 2 3
3 80 0
3sin 4sin 1lim lim
5 5 5x xx x x x
x x x x
2) 30
1 1lim
1 1xxx
.
4 Ths.NGUYN QUC TIN
Khi 0x ta c 12 11 1 (1 ) 1
2x x x ; 3
13 11 1 (1 ) 1
3x x x
Suy ra 3
1 1 321 1
xx
. Vy
30
1 1 3lim
21 1xxx
3) 0
sinlimx
tgx xx
Khi 0x , ta c:
sin 2 khi 0tgx x x x xx x
. Do 0
sinlim 2x
tgx xx
4) Tnh 3
30
sin sinlimx
tgx x xx
.
Ta c
2
3
1.sin (1 cos ) 12sincos 1 2
x xx xtgx x xx
khi 0x
Do 3 3 3 31 3sin sin2 2
tgx x x x x x khi 0x
Suy ra 3
3
3 3
3sin sin 32
2
xtgx x xx x
khi 0x
Vy 0
3
3
sin sin 3lim2x x
tgx x xx
1.3 Hm s lin tc
1.3.1 Cc nh ngha Hm s ( )y f x c gi l lin tc ti ox D nu 0
0
lim ( ) ( )x x
f x f x
. Khi 0x gi l
im lin tc ca hm ( )f x .
Hm s ( )y f x c gi l lin tc trn ( , )a b nu ( )f x lin tc ti mi im thuc ( , )a b .
Hm s ( )y f x c gi l lin tc bn tri (bn phi) 0x D nu
00
lim ( ) ( )x x
f x f x
( 00
lim ( ) ( )x x
f x f x
).
Hm ( )f x c gi l lin tc trn [ , ]a b nu ( )f x lin tc trn ( , )a b v lin tc bn phi ti
a, bn tri ti b.
1.3.2 Tnh cht ca hm s lin tc Gi s ( ), ( )f x g x l hai hm lin tc trn [ , ]a b . Khi :
5 Ths.NGUYN QUC TIN
i) ( ) ( )f x g x v ( ) ( )f x g x lin tc trn [ , ]a b , nu ( ) 0g x th ( )( )
f x
g x lin tc trn [ , ]a b .
ii) ( )f x lin tc trn [ , ]a b .
iii) Nu ( )u x lin tc ti 0x v ( )f u lin tc ti 0 0( )u u x th hm 0 ( )f u x lin tc ti 0x .
iv) ( )f x lin tc trn [ , ]a b th t gi tr ln nht, gi tr b nht trn on .
1.3.3 im gin on Nu ( )f x khng lin tc ti 0x D th ta ni ( )f x gin on ti 0x v im 0x gi l im
gin on.
Hm ( )f x gin on tai 0x nhng tn ti gii hn ca f(x) ti 0 0 , x x th 0x c gi l im
gin on loi 1. Cc im gin on khc gi l im gin on loi 2.
V d. Xt tnh lin tc ca hm
1)1, 0
( ) sin 2, 0
2
xf x x
x
Ta c
0 0
sin 2lim ( ) lim (0) 12x x
xf x f
x .
Vy ( )f x gin on ti 0x ,v 0x l im gin on loi 1
2)1 , 0
( )-1 , 0
x xf x
x x
Hm s gin on ti 0x v
0 0lim ( ) 1, lim ( ) 1x x
f x f x
nn 0x l im gin on loi 1
3) 2 3( )2
xf x
x
, c im gin on ti 0 2x
Ta c 2
lim ( )x
f x
v 2
lim ( )x
f x
Suy ra 0 2x l im gin on loi 2.
BI TP CHNG I
Cu 1. Tm min xc nh ca hm s
6 Ths.NGUYN QUC TIN
a) 2ln 1y x b) 1arctan1
yx
c) 21
1x
x x
d) 2 1x xe c)
2
sin2 3x
x x f)
2
12
xx x
Cu 2. Tnh gii hn ca cc dy s sau:
a) 2lim ( )n
n n n
; ds 12
b)4
2lim 1nn n n
n
;ds 1
c) 3 4lim2 7
n n
n nn
;ds 0 d) 1 1 1lim ...1.2 2.3 .( 1)n n n
Cu 3. Tnh gii hn ca cc hm s sau:
a) 2
2
2 1lim2 3x
x x xx x
b) 2
21
1lim4 3x
xx x
d) 3
21
1lim1x
xx
e)4 4
3 3limx ax ax a
f) 2lim( 2 )x
x x x
g) 2lim(2 2 )x
x x x
a) 4
2 1 3lim4x
xx
b) 242lim
3 4xx
x x
d)
3
41
1lim1x
xx
Cu 4. Tnh gii hn ca cc hm s sau:
a) 2
20
(1 cos )limsin tanx
xx x x ; ds 1/4 b) 20
1 cos 2limsinx
xx
; ds 1
c)0
sin 3limln(2 1)x
xx
; ds 3/2 d) 30sinlim
x
tgx xx ; ds
Cu 5. Tnh gii hn ca cc hm s sau: a) cot
0lim(s in cos ) xx
x x
; ds e b) 0
lim lnx
x x
; ds 0
c) lim xx
xe
; ds 0 d)1
2( 1)
1lim xx
x
; ds e
e) 3 5
2 60
sin tanlim3 9x
x x xx x x
;ds 1/3 f) 21
0lim(cos ) xx
x
Cu 6. Tm a cc hm s sau lin tc trn tp xc nh ca chng.
a) 21 ( 0)
( 0)
xy x
a x
b) 2
2
1 cos ( 0)
2 ( 0)2
x xxy
a x x
c) 2ln ( 0)
( 0)x x x
ya x
Cu 7. Tm cc im gin on ca hm s v chng thuc loi no
a) 12 5xyx
b) 2 2
2x xy
x
c) 0 01 0
xy
x
sin ( 0)
( 0)
x xxy
a x
7 Ths.NGUYN QUC TIN
2 CHNG 2. PHP TNH VI PHN HM MT BIN 2.1 o hm
2.1.1 o hm ti mt im
Cho hm s ( )y f x xc nh ti 0x v ti ln cn 0x . Khi nu t s 00
( ) ( )f x f x
x x
c gii hn khi 0x x th ta ni ( )f x kh vi ti 0x hay ( )f x c o hm ti 0x v gii
hn c gi l o hm ca ( )f x ti 0x . K hiu l 0'( )f x hay 0'( )y x .Vy
0
00
( ) ( )'( ) lim
f x f xf x
x xx x
.
Nu t
00
0 0
x x xx x x
x x x
Lc
0 00 0
( ) ( )'( ) lim
x
f x x f xf x
x
Hm s ( )y f x c gi l c o hm trn khong ( , )a b nu n c o hm ti mi im 0 ( , )x a b . Khi