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