-
7/25/2019 Gi i h n Hm s (Ph n 2)
1/33
GiI HN HM S
(phn 2)
V cng b v cng ln
-
7/25/2019 Gi i h n Hm s (Ph n 2)
2/33
NH NGHA
(x) l v cng b khi x xonu gi !"# c$%(x) "&! b khi x gn
xo'
0
lim ( ) 0x x
x
=
(x) l v cng ln khi x xonu gi !"# c$% (x) "&! ln khi x gn
xo'
0
lim ( )x x
x
= +
-
7/25/2019 Gi i h n Hm s (Ph n 2)
3/33
*+ ,-
01 / 0, lim 0
xx
> =
2 / 0, limx
x
+> = +
15 / lim ln 0
xx
=
3 / lim lnx
x+ = +
04 / lim ln
xx
+=
x. / 0l *1 khi x0
x. / 0l *13 khi x4
lnx l *1 khi x5
l *13 khi x 4. 0
-
7/25/2019 Gi i h n Hm s (Ph n 2)
4/33
67NH 1H86 19A *: 1;NG u 2*1
1 10( ) ( ), ( )
( ) ( )( )
khi
x x x x
x
x
x
x
/
: :
:
1 1( ) ( ) ( ) ( )x x x x :
(ch^ !h%[ !MIng DMIng _u% hi>u nu 2*1 b%n Du khng !MIng
DMIng)
Ngu[\n !]c !h%[ !MIng DMIng *1
-
7/25/2019 Gi i h n Hm s (Ph n 2)
17/33
1ch !hdc hi>n
6h%[ v _u% cc !MIng DMIng !"ung gi%n
(cheng hOn xpkhi x0). Dn khi khng cfn !h%[
DMUc n%. nu hi>u !"i>! !i\u!hT v l 2 *1
!MIng DMIngkhng !h%[ _u% hi>u !"ong
!"Mng hUp n[
-
7/25/2019 Gi i h n Hm s (Ph n 2)
18/33
*7 XY
2x
21/ arctan sinx x
x
3 / ( 1)( 1) sinx
e x x + 24 / ln( 1) sinx x x +
x1x 2xx
x
x
x
x x x
2 / tan sinx x
-
7/25/2019 Gi i h n Hm s (Ph n 2)
19/33
3Mu J
5'Fhngchu[n v !"ong !MIng DMIng cI bZn'2'Fhng!h%[ !MIng DMIng
_u% hj PW ngoOi
!" hj l[ !h% ,MIng(ch^ !h%[ !MIng
DMIng cho *1. *13')
?'6+nh !"i>! !i\u !"ong !MIng DMIng !=ng hi>u
ch^ x! cho !ng cEp hj
-
7/25/2019 Gi i h n Hm s (Ph n 2)
20/33
ln(1 sin ) sinx x x+ : :
sin 1 1x xe e x : :
ln(1 sin ) ln(1 )x x x+ +: :
sin 1 sinxe x x : :
1xe x+:
*+ ,-
2 2sin x x:
m! !+nh Dng. P%i !"ong cc !MIng DMIng P%u
Fhi x 0
2 2sin x x :
-
7/25/2019 Gi i h n Hm s (Ph n 2)
21/33
*7 XY 6NG Hq
5'6Tj cc hrng PW % v p P%o cho
, 0khipx ax x :
2'@o Pnh bc cc *1
?'6+nh gii hOn
-
7/25/2019 Gi i h n Hm s (Ph n 2)
22/33
6Tj cc hrng PW % v p P%o cho , 0khipx ax x :
21 / ( ) sin( 2 )x x x = 2 2x x: 2x:
22 / ( ) sin( tan 2 )x x x =
2
tan2x x:2 2x x:
% s2. p 5
% s2. p 5
-
7/25/2019 Gi i h n Hm s (Ph n 2)
23/33
6Tj cc hrng PW % v p P%o cho , 0khipx ax x :
3 / ( ) sin tan 2x x x =
2x x: x= % s5. p 5
34 / ( ) ln 1 sin ( 1)xx e = +
3sin ( 1)xe :3( 1)xe : 3x: % s5. p ?
-
7/25/2019 Gi i h n Hm s (Ph n 2)
24/33
6Tj cc hrng PW % v p P%o cho , 0khipx ax x :
5 / ( ) 2 2x x = +
2 2
x
x= + +
2 2
x
: 1 , 12 2
a p = =
-
7/25/2019 Gi i h n Hm s (Ph n 2)
25/33
@o Pnh bc cc *1 khi x 0
2
3 2( ) sin( 2 )1 /( ) 1 3 1
x x x
x x
=
=
2
( ) 2
1
( ) ( 3 )3
x x
x x
:
:
( ) ( ( ))x o x = ((x) bc c%o hIn (x))
c 5 !hto x
c 2 !hto x
-
7/25/2019 Gi i h n Hm s (Ph n 2)
26/33
( 1)( ) arctan2 /( ) 1x x
x x
x e
+
=
=
( ) ( 1)x x x +:
( )x x :
x:
(x) (x)
@o Pnh bc cc *1 khi x 0
-
7/25/2019 Gi i h n Hm s (Ph n 2)
27/33
@o Pnh bc cc *1 khi x 4
13 / ( ) , ( )ln x
xx x
x e
= =
1limln
x
x
e
x x+= ( )lim( )xxx
+
2lim
ln
x
x
x ex x+
= = +
(x) bc c%o hIn (x)
-
7/25/2019 Gi i h n Hm s (Ph n 2)
28/33
6+nh gii hOn
( )22
0
1 tan 3limx
x x x
x=
( )2 2
3 tan 3
20
1lim
x x x x
x
e e
x
=( )
2tan 3
3 20
limsin
x x x
x
e e
x x
+
( )2 22
0
3lim 2x
x x
x
= =
-
7/25/2019 Gi i h n Hm s (Ph n 2)
29/33
5'6+ch cc *13 l *13'
2'c 0. (x) l *13 c(x) l *13'
?' (x) b# chn !"ong ln cn xo.(x) l *1 khi x xo
(x) 4 (x) l *13 khi x xo'
6+nh ch&! v cng ln
-
7/25/2019 Gi i h n Hm s (Ph n 2)
30/33
@ @BNH C1 1B1 *: 1;NG 3wN
(x) v (x) l 2 *13 khi x xo
. DE!
0
( )lim
( )x x
xK
x
=
5'F 4 . (x) l *1 bc c%o hIn (x)'
2'F0. K (x) v (x) DLng bc'F 5K (x) v (x) !MIng DMIngK (x)
(x)
-
7/25/2019 Gi i h n Hm s (Ph n 2)
31/33
5'1h^ DMUc !h%[ !MIng DMIng _u% !+ch cc *13
2'Ngu[\n !]c ng]! b` *13 bc !h&pK !=ng cc *13
khc c&p !MIng DMIng vi *13 bc c%o nh&!
?' (x) 5(x). khi xxo.0
lim ( ) 0x x
f x a =
( ) ( )f x x ( )a x: 1( )a x:
a'Ngu[\n !]c !h%[ !MIng DMIng !"ong !+nh gii hOnKgiWng *1
Ngu[\n !]c !h%[ !h *13
' (x)b# chn !"ong ln cn xo. (x) l *13
khi x xo(x) 4 (x) (x) khi x xo'
-
7/25/2019 Gi i h n Hm s (Ph n 2)
32/33
*7 XY
lnp xx x a= =
5'Fhi x 4. j / n /0Kxjl *13 bc c%o hIn xn'
2'Fhi x 4. p / 0. / 0. % / 5K
,ln. sin ln3 Khi +x x x x + :
-
7/25/2019 Gi i h n Hm s (Ph n 2)
33/33
/2
1/ limx
xx
xe
x e+
+
/2
limx
x
x
xe
e+
=
/2lim
xx
x
e+
=
/22lim 2 0x
x
x
e+= =
/2
2 / limx
xx
xe
x e +
/2
limx
x
xe
x=
/2lim 0xx
e
= =
