7.3. H`am liˆen tu
.
c 47
nguyˆen) sao cho ta
.
id´o h `a m b ˘a
`
ng h˘a
`
ng sˆo
´
. Do vˆa
.
y n´o liˆen tu
.
cta
.
i x
0
.
Nˆe
´
u x
0
= n l`a sˆo
´
nguyˆen th`ı [n − 0] = n −1, [n +0]=n.T`u
.
d
´o suy
r˘a

1) f(x)=
x
2
x
, 2) f(x)=e
−
1
x
, 3) f(x)=



x nˆe
´
u x  1
lnx nˆe
´
u x>1.
Gia
’
i
1) H`am f(x)=x nˆe
´
u x = 0 v`a khˆong x´ac d
i
.
nh khi x =0. V`ı ∀a
ta c´o lim
x→a
x = a nˆen khi a =0:

f(x) = lim
x→0
x =0.
2) H`am f(x)=e
−
1
x
l`a h`am so
.
cˆa
´
pv`ı n´o l`a ho
.
.
pcu
’
a c´ac h`am
y = −x
−1
v`a f = e
y
.Hiˆe
’
n nhiˆen l`a h`am f(x) x´ac di
.
nh ∀x =0v`a
do d´o n´o liˆen tu
.
c ∀x =0. V`ı h`am f(x) x´ac di
.

x
n

= −∞ nˆen lim
x→∞
e
−
1
x
n
=0. T`u
.
d´o suy r˘a
`
ng lim
x→0+0
e
−
1
x
=0.
Bˆay gi`o
.
ta x´et d˜ay vˆo c`ung b´e bˆa
´
tk`y(x

n
) sao cho x


.
vˆa
.
y gi´o
.
iha
.
n bˆen tr´ai cu
’
a h`am f(x)ta
.
id
iˆe
’
m x = 0 khˆong tˆo
`
n
ta
.
idod´odiˆe
’
m x =0l`adiˆe
’
m gi´an doa
.
nkiˆe
’
uII.
48 Chu
.

adiˆe
’
m x = a khˆong ch´u
.
ad
iˆe
’
m
x =1nˆe
´
u ε<|a − 1|. Trong ε-lˆan cˆa
.
n n`ay h`am f(x) ho˘a
.
ctr`ung v´o
.
i
h`am ϕ(x)=x nˆe
´
u a<1 ho˘a
.
ctr`ung v´o
.
i h`am ϕ(x)=lnx nˆe
´
u a>1.
V`ı c´ac h`am so
.
cˆa
´

’
l`am
viˆe
.
cd
´o ta cˆa
`
n t´ınh c´ac gi´o
.
iha
.
nmˆo
.
t ph´ıa cu
’
a f(x)ta
.
id
iˆe
’
m x = a =1.
Ta c´o
f(1 + 0) = lim
x→1+0
f(x) = lim
x→1+0
lnx =0,
f(1 − 0) = lim
x→1−0
f(x) = lim

iˆe
’
m gi´an doa
.
ncu
’
a h`am
1. f(x)=
|2x − 3|
2x − 3
(DS. H`am x´ac di
.
nh v`a liˆen tu
.
c ∀x =
3
2
;ta
.
i
x
0
=
3
2
h`am c´o gi´an doa
.
nkiˆe
’
uI)

0
nˆe
´
u:
1) f(x)=



4 · 3
x
nˆe
´
u x<0
2a + x khi x  0.
(DS. H`am f liˆen tu
.
c ∀x ∈ R nˆe
´
u a =2)
7.3. H`am liˆen tu
.
c 49
2) f(x)=



x sin
1
x
,x=0;

|sin x|
sin x
(DS. H`am c´o gi´an doa
.
nta
.
i x = kπ, k ∈ Z v`ı:
f(x)=



1nˆe
´
u sin x>0
−1nˆe
´
u sin x<0)
5. f(x)=E(x) − E(−x)
(D
S. H`am c´o gi´an doa
.
nkhu
.
’
du
.
o
.
.
cta

∞)
T`ım diˆe
’
m gi´an doa
.
n v`a t´ınh bu
.
´o
.
c nha
’
ycu
’
a c´ac h`am:
7. f(x)=x +
x +2
|x +2|
(D
S. x = −2l`adiˆe
’
m gi´an doa
.
nkiˆe
’
uI,δ(−2) = 2)
50 Chu
.
o
.
ng 7. Gi´o

’
sung c´ac h`am sau dˆay ta
.
idiˆe
’
m x =0dˆe
’
ch´ung tro
.
’
th`anh
liˆen tu
.
c
9. f(x)=
tgx
x
(DS. f(0) = 1)
10. f(x)=
√
1+x − 1
x
(D
S. f(0) =
1
2
)
11. f(x)=
sin
2

.
cgo
.
il`abu
.
´o
.
c nha
’
y cu
’
a h`am f(x)ta
.
idiˆe
’
m x
0
.T`ım diˆe
’
m gi´an doa
.
n
v`a bu
.
´o
.
c nha
’
y cu
’









2
√
x nˆe
´
u0 x  1;
4 − 2x nˆe
´
u1<x 2, 5;
2x − 7nˆe
´
u2, 5  x<+∞.
(D
S. x
0
=2, 5l`adiˆe
’
m gi´an doa
.
nkiˆe
’
uI;d = −1)
3) f(x)=

uI,d = −4)
7.4. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am nhiˆe
`
ubiˆe
´
n 51
7.4 Gi´o
.
iha
.
nv`aliˆen tu
.
ccu
’
ah`am nhiˆe
`
u
biˆe
´
n
1. Gia
’

am˘a
.
t ph˘a
’
ng v`a x → x
0
, y → y
0
,
khi d´o d iˆe
’
m M(x, y) → M
0
(x
0
,y
0
). Diˆe
`
u n`ay tu
.
o
.
ng du
.
o
.
ng v´o
.
i khoa

1/2
.
Ta c´o c´ac d
i
.
nh ngh˜ıa sau dˆay:
i) Di
.
nh ngh˜ıa gi´o
.
iha
.
n (theo Cauchy)
Sˆo
´
b d
u
.
o
.
.
cgo
.
i l`a gi´o
.
iha
.
ncu
’
a h`am f(M) khi M → M

cgo
.
i l`a gi´o
.
iha
.
ncu
’
a h`am f(M)ta
.
idiˆe
’
m M
0
nˆe
´
udˆo
´
iv´o
.
i
d˜ay diˆe
’
m {M
n
} bˆa
´
tk`yhˆo
.
itu

K´yhiˆe
.
u:
i) lim
M→M
0
f(M)=b, ho˘a
.
c
ii) lim
x → x
0
y → y
0
f(x, y)=b
Hai di
.
nh ngh˜ıa gi´o
.
iha
.
ntrˆendˆay tu
.
o
.
ng du
.
o
.
ng v´o

0
.Dod´onˆe
´
u M → M
0
theo
c´ac hu
.
´o
.
ng kh´ac nhau m`a f(M)dˆa
`
nd
ˆe
´
n c´ac gi´a tri
.
kh´ac nhau th`ı khi
M → M
0
h`am f(M) khˆong c´o gi´o
.
iha
.
n.
52 Chu
.
o
.
ng 7. Gi´o

´
iv´o
.
i h`am nhiˆe
`
ubiˆe
´
n, c`ung v´o
.
i gi´o
.
iha
.
n thˆong thu
.
`o
.
ng d
˜anˆeuo
.
’
trˆen (gi´o
.
iha
.
nk´ep !), ngu
.
`o
.
i ta c`on x´et gi´o

2
}
c´o thˆe
’
tr `u
.
ra ch´ınh c´ac diˆe
’
m x = x
0
, y = y
0
. Khi cˆo
´
di
.
nh mˆo
.
t gi´a tri
.
y th`ı h`am f(x, y) tro
.
’
th`anh h`am mˆo
.
tbiˆe
´
n. Gia
’
su

i gi´o
.
iha
.
n
lim
x→x
0
y cˆo
´
di
.
nh
f(x, y)=ϕ(y).
Tiˆe
´
p theo, gia
’
su
.
’
lim
y→y
0
ϕ(y)=b tˆo
`
nta
.
i. Khi d´o ngu
.

lim
y→y
0
lim
x→x
0
f(x, y)=b,
trong d´o gi´o
.
iha
.
n lim
x→x
0
y cˆo
´
di
.
nh
0<|y−y
0
|<d
2
f(x, y)go
.
i l`a gi´o
.
iha
.
n trong. Tu

.
n
lim
y→y
0
x cˆo
´
di
.
nh
0<|x−x
0
|<d
1
f(x, y)
l`a gi´o
.
iha
.
n trong.
Mˆo
´
i quan hˆe
.
gi˜u
.
a gi´o
.
iha
.

`
ubiˆe
´
n 53
Gia
’
su
.
’
ta
.
id
iˆe
’
m M
0
(x
0
,y
0
) gi´o
.
iha
.
nk´ep v`a c´ac gi´o
.
iha
.
n trong cu
’

0
f(x, y) = lim
y→y
0
lim
x→x
0
= lim
x→x
0
y→y
0
f(x, y).
T`u
.
d
i
.
nh l´y n`ay ta thˆa
´
yr˘a
`
ng viˆe
.
c thay dˆo
’
ith´u
.
tu
.

.
nh l´y vˆe
`
c´ac t´ınh chˆa
´
t
sˆo
´
ho
.
ccu
’
a gi´o
.
iha
.
ntu
.
o
.
ng tu
.
.
c´ac di
.
nh l´yvˆe
`
gi´o
.
iha

n.
H`am u = f(M)d
u
.
o
.
.
cgo
.
il`aliˆen tu
.
c ta
.
id
iˆe
’
m M
0
nˆe
´
u:
i) f(M) x´ac di
.
nh ta
.
ich´ınh diˆe
’
m M
0
c˜ung nhu

).
Su
.
.
liˆen tu
.
cv`u
.
adu
.
o
.
.
cdi
.
nh ngh˜ıa go
.
i l`a su
.
.
liˆen tu
.
c theo tˆa
.
pho
.
.
p
biˆe
´

o
.
.
cgo
.
il`ad
iˆe
’
m gi´an doa
.
n cu
’
a h`am f(M)nˆe
´
udˆo
´
iv´o
.
i
d
iˆe
’
m M
0
c´o ´ıt nhˆa
´
tmˆo
.
t trong ba diˆe
`

l`a ca
’
mˆo
.
tdu
.
`o
.
ng (du
.
`o
.
ng gi´an doa
.
n).
Nˆe
´
u h`am f(x, y)liˆen tu
.
cta
.
id
iˆe
’
m M
0
(x
0
,y
0

i l`a khˆong
d´ung.
C˜ung nhu
.
dˆo
´
iv´o
.
i h`am mˆo
.
tbiˆe
´
n, tˆo
’
ng, hiˆe
.
u v`a t´ıch c´ac h`am liˆen
tu
.
c hai biˆe
´
nta
.
id
iˆe
’
m M
0
l`a h`am liˆen tu
.

0
h`am
54 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo
´
mˆa
˜
usˆo
´
kh´ac 0. Ngo`ai ra, di
.
nh l´y vˆe
`
t´ınh liˆen tu
.
ccu
’
a h`am ho
.

ba
’
n liˆen quan d
ˆe
´
n gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am ba biˆe
´
n,
C
´
AC V
´
IDU
.
V´ı du
.
1. Ch´u
.
ng minh r˘a
`
ng h`am
f(x, y)=(x + y) sin

i h`am mˆo
.
t
biˆe
´
n) ta cˆa
`
nch´u
.
ng minh r˘a
`
ng
lim
x→0
y→0
f(x, y)=0.
Ta´apdu
.
ng d
i
.
nh ngh˜ıa gi´o
.
iha
.
n theo Cauchy. Ta cho sˆo
´
ε>0t`uy
´yv`ad
˘a




 |x| + |y| < 2δ = ε.
Diˆe
`
ud´och´u
.
ng to
’
r˘a
`
ng
lim
x→0
y→0
f(x, y)=0.
V´ı du
.
2. T´ınh c´ac gi´o
.
iha
.
n sau d
ˆay:
1) lim
x→0
y→2

1+xy

.
7.4. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am nhiˆe
`
ubiˆe
´
n 55
Gia
’
i. 1) Ta biˆe
’
udiˆe
˜
n h`am du
.
´o
.
idˆa
´
u gi´o
.
iha
.

xy
= lim
t→0

1+t

1
t
= e.
Tiˆe
´
p theo v`ı lim
x→0
y→2
2
x + y
= 2 (theo di
.
nh l´y thˆong thu
.
`o
.
ng vˆe
`
gi´o
.
iha
.
n
cu

’
ng c´ach
gi˜u
.
a hai diˆe
’
m M v`a M
0
b˘a
`
ng
ρ =

x
2
+(y − 2)
2
.
Do d´o
lim
x→0
y→2
f(x, y) = lim
ρ→0

ρ
2
+1− 1
ρ
2

.
.
ctac´ox = ρ cos ϕ, y = ρ sin ϕ.Tac´o
x
4
+ y
4
x
2
+ y
2
=
ρ
4
(cos
4
ϕ + sin
4
ϕ)
ρ
2
(cos
2
ϕ + sin
2
ϕ)
= ρ
2
(cos
4

o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo
´
V´ı du
.
3. 1) Ch´u
.
ng minh r˘a
`
ng h`am
f
1
(x, y)=
x −y
x + y
khˆong c´o gi´o
.
iha
.
nta
.

.
i ngoa
.
itr`u
.
du
.
`o
.
ng th˘a
’
ng
x + y =0. Tach´u
.
ng minh r˘a
`
ng h`am khˆong c´o gi´o
.
iha
.
nta
.
i(0, 0). Ta
lˆa
´
y hai d˜ay d
iˆe
’
mhˆo
.

.
.
c
lim
n→∞
f
1
(M
n
) = lim
n→∞
1
n
−0
1
n
+0
=1;
lim
n→∞
f
1
(M

n
) = lim
n→∞
0 −
1
n

’
a h`am khˆong c´o c`ung gi´o
.
iha
.
n. Do d
´o
theo di
.
nh ngh˜ıa h`am khˆong c´o gi´o
.
iha
.
nta
.
i(0, 0).
2) Gia
’
su
.
’
diˆe
’
m M(x, y)dˆa
`
ndˆe
´
ndiˆe
’
m(0, 0) theo du

x
2
=
k
1+k
2
·
7.4. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am nhiˆe
`
ubiˆe
´
n 57
Nhu
.
vˆa
.
y khi dˆa
`
nd
ˆe
´
ndiˆe

.
n kh´ac nhau, t´u
.
c l`a h`am d˜a cho khˆong c´o gi´o
.
iha
.
nta
.
i(0, 0). 
V´ı d u
.
4. Kha
’
o s´at t´ınh liˆen tu
.
ccu
’
a c´ac h`am
1) f(x, y)=
x
2
+2xy +5
y
2
− 2x +1
2) f(x, y)=
1
x
2

mcu
’
am˘a
.
t ph˘a
’
ng R
2
m`a to
.
adˆo
.
cu
’
ach´ung tho
’
a m˜an phu
.
o
.
ng tr`ınh
y
2
−2x +1=0. D´ol`aphu
.
o
.
ng tr`ınh du
.
`o

.
n
-d
´ol`adu
.
`o
.
ng gi´an doa
.
ncu
’
a h`am. Nh˜u
.
ng diˆe
’
mcu
’
am˘a
.
t ph˘a
’
ng R
2
khˆong thuˆo
.
c parabˆon d´ol`anh˜u
.
ng diˆe
’
m liˆen tu

2
− z =0. D´o l`a phu
.
o
.
ng tr`ınh
m˘a
.
t paraboloit tr`on xoay. Trong tru
.
`o
.
ng ho
.
.
p n`ay m˘a
.
t paraboloit l`a
m˘a
.
t gi´an d
oa
.
ncu
’
a h`am.
3) V`ı tu
.
’
sˆo

3
= 0. H`am c´o gi´an doa
.
nta
.
i
nh˜u
.
ng diˆe
’
mm`ax
3
+ y
3
=0hayy = −x. Ngh˜ıa l`a h`am c´o gi´an doa
.
n
trˆen du
.
`o
.
ng th˘a
’
ng y = −x.
Gia
’
su
.
’
x

0
y
0
+ y
2
0
·
T`u
.
d´o suy ra r˘a
`
ng c´ac diˆe
’
mcu
’
adu
.
`o
.
ng th˘a
’
ng y = x (x = 0) l`a
58 Chu
.
o
.
ng 7. Gi´o
.
iha
.

+ y
3
= lim
x→0
y→0
1
x
2
− xy + y
2
=+∞
nˆen d
iˆe
’
m O(0, 0) l`a diˆe
’
m gi´an doa
.
nvˆoc`ung.
B
`
AI T
ˆ
A
.
P
Trong c´ac b`ai to´an sau d
ˆay (1-10) h˜ay t`ım miˆe
`
n x´ac di

2
 a
2
)
4. w =
1

x
2
+ y
2
− a
2
.(DS. x
2
+ y
2
>a
2
)
5. w =

1 −
x
2
a
2
−
y
2

√
xy.(DS. Hai nu
.
’
a b˘ang vˆo ha
.
n th˘a
’
ng d´u
.
ng
{0  x  2, 0  y<+∞} v`a {−2  x  0, −∞ <y 0})
8. w =

x
2
+ y
2
− 1 + ln(4 − x
2
− y
2
).
(DS. V`anh tr`on 1  x
2
+ y
2
< 4)
9. w =


n trong cu
’
amˆa
.
t parab oloid z = x
2
+ y
2
−1).
Trong c´ac b`ai to´an sau d
ˆay (11-18) h˜ay t´ınh c´ac gi´o
.
iha
.
ncu
’
a h`am
7.4. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am nhiˆe
`
ubiˆe
´
n 59

x
2
+ y
2
+1− 1
.(DS. 2)
Chı
’
dˆa
˜
n. Su
.
’
du
.
ng khoa
’
ng c´ach ρ =

x
2
+ y
2
ho˘a
.
c nhˆan - chia
v´o
.
id
a

)
16. lim
x→0
y→0
x
2
y
x
2
+ y
2
.(DS. 0)
17. lim
x→0
y→5

(x
2
+(y − 5)
2
+1− 1
x
2
+(y − 5)
2
.(DS.
1
2
)
18. lim

8.1.2 D
-
a
.
o h`am cˆa
´
pcao 62
8.2 Viphˆan 75
8.2.1 Vi phˆan cˆa
´
p1 75
8.2.2 Vi phˆan cˆa
´
pcao 77
8.3 C´ac d
i
.
nh l´y co
.
ba
’
nvˆe
`
h`am kha
’
vi. Quy
t˘a
´
c l’Hospital. Cˆong th´u
.

.
o h`am 61
8.1 D
-
a
.
oh`am
8.1.1 D
-
a
.
o h`am cˆa
´
p1
Gia
’
su
.
’
h`am y = f(x)x´acdi
.
nh trong δ-lˆan cˆa
.
ncu
’
adiˆe
’
m x
0
(U(x

´
gia ∆x = x − x
0
cu
’
adˆo
´
isˆo
´
.
Theo d
i
.
nh ngh˜ıa: Nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
nh˜u
.
uha
.
n
lim
∆x→0

v`a du
.
o
.
.
cchı
’
bo
.
’
imˆo
.
t trong c´ac k´yhiˆe
.
u:
lim
∆x→0
f(x
0
+∆x) − f(x
0
)
∆x
≡
dy
dx
≡
d
dx
f(x) ≡ f

v`a
f

−
(x
0
)=f

(x
0
− 0) = lim
∆x→0
∆x<0
∆y
∆x
= lim
∆x→0−0
∆y
∆x
du
.
o
.
.
cgo
.
il`ada
.
oh`ambˆen pha
’

.
iha
.
nmˆo
.
tph´ıa ta c´o:
D
-
i
.
nh l´y 8.1.1. H`am y = f(x) c´o d
a
.
o h`am ta
.
idiˆe
’
m x khi v`a chı
’
khi
c´ac da
.
o h`am mˆo
.
tph´ıa tˆo
`
nta
.
iv`ab˘a
`


(x)tˆo
`
nta
.
i v`a liˆen tu
.
c. Nˆe
´
u h`am f(x) kha
’
vi th`ı n´o liˆen tu
.
c. D
iˆe
`
u kh˘a
’
ng di
.
nh ngu
.
o
.
.
cla
.
i l`a khˆong d´ung.
62 Chu
.

.
o h`am bˆa
.
c nhˆa
´
t).
Da
.
o h`am cu
’
a f

(x)du
.
o
.
.
cgo
.
il`ada
.
o h`am cˆa
´
p hai (hay da
.
o h`am th´u
.
hai)cu
’
a h`am f(x)v`adu

.
o h`am th´u
.
ba)cu
’
a h`am f(x)
v`a d
u
.
o
.
.
ck´yhiˆe
.
u y

hay f

(x) (hay y
(3)
, f
(3)
(x) v.v
Ta c´o ba
’
ng da
.
o h`am cu
’
a c´ac h`am so

x
lnaa
x
(lna)
n
lnx
1
x
(−1)
n−1
(n − 1)!
1
x
n
, x>0
log
a
x
1
xlna
(−1)
n−1
(n − 1)!
1
x
n
lna
, x>0
sin x cos x sin
