Bài tập toán cao cấp part 3 ppsx

7.2. Gi´o
.
iha
.
n h`am mˆo
.
tbiˆe
´
n 31
v`a c´ac hˆe
.
qua
’
cu
’
a (7.13)
lim
x→∞

1+
1
x

x
= e, (7.14)
lim
x→0
log
a
(1 + x)
x

=
1
lna
, 0 <a=1, (7.15)
lim
x→0
a
x
− 1
x
=lna, 0 <a=1. (7.16)
C
´
AC V
´
IDU
.
V´ı du
.
1. Su
.
’
du
.
ng (ε −δ)-d
i
.
nh ngh˜ıa gi´o
.
iha

.
nd
ˆe
’
ch´u
.
ng minh r˘a
`
ng
lim
x→−3
x
2
=9.
Gia
’
i. Ta cˆa
`
nch´u
.
ng minh r˘a
`
ng ∀ε>0, ∃δ>0 sao cho v´o
.
i
|x +3| <δth`ı ta c´o |x
2
− 9| <ε.
Ta cˆa
`

nu
.
´o
.
clu
.
o
.
.
ng hiˆe
.
u |x
2
− 9|. ta c´o
|x
2
− 9| = |x −3||x +3|.
Do th`u
.
asˆo
´
|x −3| khˆong bi
.
ch˘a
.
n trˆen to`an tru
.
csˆo
´
nˆen dˆe

’
u
.
´o
.
clu
.
o
.
.
ng
t´ıch do
.
n gia
’
nho
.
n ta tr´ıch ra 1 - lˆan cˆa
.
ncu
’
adiˆe
’
m a = −3t´u
.
cl`a
khoa
’
ng (−4; −2). V´o
.

imo
.
i x ∈ (−4; −2) ta c´o |x − 3| < 7v`adod´o
|x
2
− 9| < 7|x +3|.
V`ı δ-lˆan cˆa
.
ndiˆe
’
m a = −3[t´u
.
c l`a khoa
’
ng (−3 − δ; −3+δ)] khˆong
d
u
.
o
.
.
cvu
.
o
.
.
t ra kho
’
i ranh gi´o
.

icu
’
a 1-lˆan cˆa
.
n nˆen ta lˆa
´
y δ = min

1,
ε
7

.
Khi d
´ov´o
.
i0< |x +3| <δ⇒|x
2
− 9| <ε. Do vˆa
.
y lim
x→−3
x
2
=9. 
V´ı du
.
2. Ch´u
.
ng minh r˘a

`
ng lim
x→2
√
11 − x =3.
Gia
’
i. Gia
’
su
.
’
ε>0 l`a sˆo
´
du
.
o
.
ng cho tru
.
´o
.
cb´e bao nhiˆeu t`uy ´y. Ta
x´et bˆa
´
tphu
.
o
.
ng tr`ınh

|
√
11 − x − 3| <ε. (7.17)
32 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo
´
Ta c´o
(7.17) ⇔−ε<
√
11 − x − 3 <ε⇔



√
11 − x − 3 > −ε
√
11 − x − 3 <ε
⇔




x − 11 < −(3 − ε)
2
x − 11 > −(3 + ε)
2
⇔



x − 2 < 6ε − ε
3
x − 2 > −(6ε + ε
2
).
V`ı6ε − ε
2
< |−(6ε + ε)
2
| =6ε + ε
2
nˆen ta c´o thˆe
’
lˆa
´
y δ(ε) l`a sˆo
´
δ  6ε − ε
2
.V´o

.
isˆo
´
δ d´o ta thˆa
´
yr˘a
`
ng khi x tho
’
a m˜an bˆa
´
td˘a
’
ng th´u
.
c
0 < |x − 2| <δth`ı |
√
11 − x − 3| <εv`a
lim
x→2
√
11 − x =3. 
V´ı d u
.
3. T´ınh c´ac gi´o
.
iha
.
n

1) lim
x→2
2
x
− x
2
x − 2
(vˆo di
.
nh da
.
ng
0
0
);
2) lim
x→
π
4
cotg2x · cotg

π
4
− x

(vˆo di
.
nh da
.
ng 0 ·∞);

3) lim
x→∞

e
1
x
+
1
x

x
(vˆo di
.
nh da
.
ng 1
∞
).
Gia
’
i
1) Ta c´o
2
x
− x
2
x −2
=
2
x

−2
2
−(x
2
− 2
2
)
x − 2
=4·
2
x−2
− 1
x −2
−
x
2
− 4
x − 2
·
T`u
.
d
´o suy r˘a
`
ng
lim
x→2
2
x
− x

2
x − 2
= 4 lim
x→2
2
x−2
− 1
x − 2
− lim
x→2
x
2
−4
x − 2
= 4ln2 − 4.
2) D˘a
.
t y =
π
4
−x. Khi d´o
lim
x→
π
4
cotg2x · cotg

π
4
− x


= lim
y→0
cotg

π
2
− 2y

cotgy
= lim
y→0
sin 2y
sin y
·
cos y
cos 2y
=2.
7.2. Gi´o
.
iha
.
n h`am mˆo
.
tbiˆe
´
n 33
3) D˘a
.
t y =

1
x
. Khi d´o
lim
x→∞

e
1
x
+
1
x

x
= lim
y→0
(e
y
+ y)
1
y
= e
lim
y→0
ln
(e
y
+
y)
y

;
lim
y→0
ln(e
y
+ y)
y
= lim
y→0
ln[1 + (e
y
+ y −1)]
e
y
+ y − 1
·
e
y
+ y − 1
y
= lim
t→0
ln(1 + t)
t
· lim
y→0

1+
e
y

− 1
y

=2.
T`u
.
d´o suy r˘a
`
ng
lim
y→0

e
y
+ y

1
y
= e
2
. 
V´ı du
.
4. Ch´u
.
ng to
’
r˘a
`
ng h`am f( x) = sin

1
x
khˆong c´o gi´o
.
iha
.
n khi
x → 0.
Gia
’
i. Ta lu
.
u´ymˆe
.
nh d
ˆe
`
phu
’
di
.
nh dˆo
´
iv´o
.
id
i
.
nh ngh˜ıa gi´o
.

iha
.
n:
lim
x→a
f(x) = A ⇔∃ε
0
> 0 ∀δ>0 ∃x
δ
(0 < |x
δ
− a| <δ)
→|f(x
0
) − A|  ε
0
.
Nˆe
´
u A =0talˆa
´
y ε
0
=
1
2
v`a x
k
=
2

π
2
+2kπ
. Khi d´o ∀δ>0,
∃k ∈ N :0<x
k
<δv`a
|f(x
k
) − 0| = |f(x
k
)| =1>ε
0
v`a nhu
.
vˆa
.
y A = 0 khˆong pha
’
i l`a gi´o
.
iha
.
ncu
’
ah`amd
˜a cho khi x → 0.
Nˆe
´
u A = 0 th`ı ta lˆa

´
y ε
0
=
|A|
2
v`a x
k
=
1
2kπ
. Khi d´o ∀δ>0,
∃k ∈ N :0<x
k
<δth`ı |f(x
k
) − A| = |A| >ε.Nhu
.
vˆa
.
ymo
.
isˆo
´
A =0d
ˆe
`
u khˆong l`a gi´o
.
iha

.
ncu
’
a h`am sin
1
x
khi x → 0. 
V´ı du
.
5. H`am Dirichlet D( x):
D(x)=



1nˆe
´
u x ∈ Q,
0nˆe
´
u x ∈ R \Q
34 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.

ccu
’
a h`am sˆo
´
khˆong c´o gi´o
.
iha
.
nta
.
i ∀a ∈ R.
Gia
’
i. Ta ch´u
.
ng minh r˘a
`
ng ta
.
imo
.
idiˆe
’
m a ∈ R h`am D(x) khˆong
tho
’
a m˜an D
i
.
nh l´y 2. Dˆe

’
l`am viˆe
.
cd´o, ta chı
’
cˆa
`
nchı
’
ra hai d˜ay (a
n
)v`a
(a

n
)c`ung hˆo
.
itu
.
dˆe
´
n a sao cho lim
n→∞
D(a
n
) = lim
n→∞
D(a

n

).
D
ˆa
`
u tiˆen ta x´et d˜ay c´ac diˆe
’
mh˜u
.
uty
’
(a
n
)hˆo
.
itu
.
dˆe
´
n a.Tac´o
D(a
n
)=1∀n v`a do d´o lim
n→∞
D(a
n
) = 1. Bˆay gi`o
.
ta x´et d˜ay (a

n

)-
d˜ay c´ac d
iˆe
’
mvˆoty
’
hˆo
.
itu
.
dˆe
´
n a.Tac´oD(a

n
)=0∀n v`a do vˆa
.
y
lim
n→∞
D(a

n
)=0.
Nhu
.
vˆa
.
y lim
n→∞

D(a
n
) = lim
n→∞
D(a

n
). T`u
.
d
´o suy ra r˘a
`
ng ta
.
idiˆe
’
m a
h`am D(x) khˆong c´o gi´o
.
iha
.
n.
V´ı d u
.
6. Gia
’
su
.
’
lim

x→a
f(x)=b, lim
x→a
g(x)=+∞.Ch´u
.
ng minh r˘a
`
ng
lim
x→a
[f(x)+g(x)] = +∞.
Gia
’
i. Ta cˆa
`
nch´u
.
ng minh r˘a
`
ng ∀M>0, ∃δ>0 sao cho ∀x :0<
|x − a| <δth`ı f(x)+g(x) >M.
V`ı lim
x→a
f(x)=b nˆen tˆo
`
nta
.
i δ
1
-lˆan cˆa

.
n U(a, δ
1
)cu
’
adiˆe
’
m a sao cho
|f(x)| <C, x= a (7.18)
trong d
´o C l`a h˘a
`
ng sˆo
´
du
.
o
.
ng n`ao d
´o .
Gia
’
su
.
’
M>0 l`a sˆo
´
cho tru
.
´o

.
ct`uy ´y. V`ı lim
x→a
g(x)=+∞ nˆen d ˆo
´
i
v´o
.
isˆo
´
M + C, ∃δ>0(δ  δ
1
) sao cho ∀x :0< |x − a| <δth`ı
g(x) >M+ C (7.19)
T`u
.
c´ac bˆa
´
td˘a
’
ng th´u
.
c (7.18) v`a(7.19) ta thu du
.
o
.
.
c l`a: v´o
.
i x tho

’
a
m˜an diˆe
`
ukiˆe
.
n0< |x −a| <δ δ
1
th`ı
f(x)+g(x)  g(x) −|f(x)| >M+ C − C = M. 
B
`
AI T
ˆ
A
.
P
7.2. Gi´o
.
iha
.
n h`am mˆo
.
tbiˆe
´
n 35
1. Su
.
’
du

.
ng d
i
.
nh ngh˜ıa gi´o
.
iha
.
n h`am sˆo
´
d
ˆe
’
ch´u
.
ng minh c´ac d
˘a
’
ng th ´u
.
c
sau d
ˆay:
1) lim
x→
π
6
sin x =
1
2

; 2) lim
x→
π
2
sin x =1;
3) lim
x→0
x sin
1
x
= 0; 4) lim
x→+∞
arctgx =
π
2
.
Chı
’
dˆa
˜
n. D`ung hˆe
.
th ´u
.
c
π
2
− arctgx<tg

π

2
− arctgx

=
1
x
)
5) lim
x→∞
x −1
3x +2
=
1
3
; 6) lim
x→+∞
log
a
x =+∞;
7) lim
x→+∞

√
x
2
+1− x

= 0; 8) lim
x→−5
x

2
+2x − 15
x +5
= −8;
9) lim
x→1
(5x
2
− 7x + 6) = 4; 10) lim
x→2
x
2
− 3x +2
x
2
+ x − 6
=
1
5
;
11) lim
x→+∞
x sin x
x
2
− 100x + 3000
=0.
2. Ch´u
.
ng minh c´ac gi´o

.
iha
.
n sau dˆay khˆong tˆo
`
nta
.
i:
1) lim
x→1
sin
1
x −1
; 2) lim
x→∞
sin x; 3) lim
x→o
2
1
x
;
4) lim
x→0
e
1
x
; 5) lim
x→∞
cos x.
Nˆe

´
utu
.
’
sˆo
´
v`a mˆa
˜
usˆo
´
cu
’
a phˆan th ´u
.
ch˜u
.
uty
’
dˆe
`
u triˆe
.
t tiˆeu ta
.
idiˆe
’
m
x = a th`ı c´o thˆe
’
gia

’
nu
.
´o
.
c phˆan th´u
.
cchox − a (= 0) mˆo
.
t ho˘a
.
cmˆo
.
t
sˆo
´
lˆa
`
n.
Su
.
’
du
.
ng phu
.
o
.
ng ph´ap gia
’

nu
.
´o
.
cd´o, h˜ay t´ınh c´ac gi´o
.
iha
.
n sau dˆay
(3-10).
3. lim
x→7
2x
2
− 11x − 21
x
2
− 9x +14
(DS.
17
5
)
4. lim
x→1
x
4
−x
3
+ x
2

− 3x +2
x
3
−x
2
− x +1
(DS. 2)
5. lim
x→1
x
4
+2x
2
−3
x
2
−3x +2
(DS. −8)
6. lim
x→1
x
m
− 1
x
n
− 1
; m, n ∈ Z (D
S.
m
n

)
36 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo
´
7. lim
x→1

1
1 − x
−
3
1 − x
3

(DS. −1)
8. lim
x→1

a

1 − x
a
−
b
1 − x
b

; a, b ∈ N (D
S.
a −b
2
)
9. lim
x→1
(x
n
− 1)(x
n−1
− 1) ···(x
n−k+1
−1)
(x − 1)(x
2
− 1) ···(x
k
− 1)
(DS. C
k
n
)

10. lim
x→a
(x
n
− a
n
) − na
n−1
(x − a)
(x −a)
2
, n ∈ N (DS.
n(n − 1)
2
a
n−1
)
Chı
’
dˆa
˜
n. Dˆo
’
ibiˆe
´
n x −a = t.
C´ac b`ai to´an sau dˆay c´o thˆe
’
du
.

avˆe
`
da
.
ng trˆen nh`o
.
ph´ep dˆo
’
ibiˆe
´
n
(11-14)
11. lim
x→1
x
p
q
− 1
x
r
s
− 1
(D
S.
ps
qr
)
12. lim
x→−1
1+

3
√
x
1+
5
√
x
(DS.
5
3
)
13. lim
x→0
3
3
√
1+x −4
4
√
1+x +1
2 − 2
√
1+x + x
(DS.
1
6
)
14. lim
x→0
n

√
1+x − 1
x
(DS.
1
n
)
Mˆo
.
t trong c´ac phu
.
o
.
ng ph´ap t´ınh gi´o
.
iha
.
ncu
’
a c´ac biˆe
’
uth´u
.
cvˆoty
’
l`a chuyˆe
’
nvˆoty
’
t`u

.
mˆa
˜
usˆo
´
lˆen tu
.
’
sˆo
´
ho˘a
.
c ngu
.
o
.
.
cla
.
i (15-26)
15. lim
x→0
√
1+x + x
2
− 1
x
(DS.
1
2

)
16. lim
x→2
√
3+x + x
2
−
√
9 − 2x + x
2
x
2
− 3x +2
(DS.
1
2
)
17. lim
x→0
5x
3
√
1+x −
3
√
1 − x
(D
S.
15
2

)
18. lim
x→0
3
√
1+3x −
3
√
1 − 2x
x + x
2
(DS. 2)
19. lim
x→∞
√
x
2
+1−
√
x
2
− 1

(DS. 0)
7.2. Gi´o
.
iha
.
n h`am mˆo
.

tbiˆe
´
n 37
20. lim
x→∞

3
√
1 − x
3
+ x

(DS. 0)
21. lim
x→+∞

√
x
2
+5x + x

(DS. +∞)
22. lim
x→−∞
√
x
2
+5x + x

(DS. −

5
2
)
23. lim
x→+∞
√
x
2
+2x − x

(DS. 1)
24. lim
x→−∞
√
x
2
+2x − x

.(DS. +∞)
25. lim
x→∞

(x +1)
2
3
− (x −1)
2
3

(D

S. 0)
26. lim
x→+∞

n

(x + a
1
)(x + a
2
) ···(x + a
n
) − x

(D
S.
a
1
+ a
2
+ ···+ a
n
n
)
Khi gia
’
i c´ac b`ai to´an sau dˆay ta thu
.
`o
.

ng su
.
’
du
.
ng hˆe
.
th ´u
.
c
lim
t→0
(1 + t)
α
− 1
t
= α (27-34)
27. lim
x→0
5
√
1+3x
4
−
√
1 − 2x
3
√
1+x −
√

1+x
(DS. −6)
28. lim
x→0
n
√
a + x −
n
√
a − x
x
, n ∈ N (DS.
2
n
a
1
n
−1
)
29. lim
x→0
√
1+3x +
3
√
1+x −
5
√
1+x −
7

√
1+x
4
√
1+2x + x −
6
√
1+x
(D
S.
313
280
)
30. lim
x→0
3
√
a
2
+ ax + x
2
−
3
√
a
2
− ax + x
2
√
a + x −

√
a −x
(D
S.
3
2
a
1
6
)
31. lim
x→0

√
1+x
2
+ x

n
−

√
1+x
2
− x

n
x
(DS. 2n)
32. lim

x→0
n
√
a + x −
n
√
a − x
x
, n ∈ N, a>0(DS.
2
n
√
a
na
)
33. lim
x→0
n
√
1+ax −
k
√
1+bx
x
, n ∈ N, a>0(DS.
ak −bn
nk
)
34. lim
x→∞


n

(1 + x
2
)(2 + x
2
) ···(n + x
2
) − x
2

(D
S.
n +1
2
)
38 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo

´
Khi t´ınh gi´o
.
iha
.
n c´ac biˆe
’
uth´u
.
clu
.
o
.
.
ng gi´ac ta thu
.
`o
.
ng su
.
’
du
.
ng cˆong
th ´u
.
cco
.
ba
’

n
lim
x→0
sin x
x
=1
c`ung v´o
.
isu
.
.
kˆe
´
tho
.
.
p c´ac phu
.
o
.
ng ph´ap t`ım gi´o
.
iha
.
nd
˜a n ˆeu o
.
’
trˆen
(35-56).

35. lim
x→∞
sin
πx
2
x
(DS. 0)
36. lim
x→∞
arctgx
2x
(DS. 0)
37. lim
x→−2
x
2
− 4
arctg(x +2)
(DS. −4)
38. lim
x→0
tgx − sin x
x
3
(DS.
1
2
)
39. lim
x→0

xcotg5x (DS.
1
5
)
40. lim
x→1
(1 − x)tg
πx
2
(DS.
2
π
)
41. lim
x→1
1 − x
2
sin πx
(DS.
2
π
)
42. lim
x→π
sin x
π
2
− x
2
(DS.

1
2π
)
43. lim
x→0
cos mx − cos nx
x
2
(DS.
1
2
(n
2
− m
2
))
44. lim
x→∞
x
2

cos
1
x
− cos
3
x

(DS. 4)
45. lim

x→0
sin(a + x) + sin(a −x) −2sin a
x
2
(DS. −sin a)
46. lim
x→0
cos(a + x) + cos(a −x) −2 cos a
1 − cos x
(D
S. −2 cos a)
47. lim
x→∞

sin
√
x
2
+1− sin
√
x
2
− 1

(DS. 0)
7.2. Gi´o
.
iha
.
n h`am mˆo

.
tbiˆe
´
n 39
48. lim
x→0
√
cos x − 1
x
2
(DS. −
1
4
)
49. lim
x→
π
2
cos
x
2
− sin
x
2
cos x
(D
S.
1
√
2

)
50. lim
x→
π
3
sin

x −
π
3

1 − 2 cos x
(D
S.
1
√
3
)
51. lim
x→
π
4
√
2 cos x − 1
1 − tg
2
x
(D
S.
1

4
)
52. lim
x→0
√
1+tgx −
√
1 − tgx
sin x
(DS. 1)
53. lim
x→0
m
√
cos αx −
m
√
cos βx
x
2
(DS.
β
2
− α
2
2m
)
54. lim
x→0
cos x −

3
√
cos x
sin
2
x
(DS. −
1
3
)
55. lim
x→0
1 − cos x
√
cos 2x
tgx
2
(DS.
3
2
)
56. lim
x→0
√
1+x sin x − cos x
sin
2
x
2
(DS. 4)

D
ˆe
’
t´ınh gi´o
.
iha
.
n lim
x→a
[f(x)]
ϕ(x)
, trong d´o
f(x) → 1, ϕ(x) →∞khi x → a ta c´o thˆe
’
biˆe
´
nd
ˆo
’
ibiˆe
’
uth´u
.
c
[f(x)]
ϕ(x)
nhu
.
sau:
lim

x→a
[f(x)]
ϕ(x)
= lim
x→a

[1 + (f(x) −1)]
1
f(x)−1

ϕ(x)[f(x)−1]
= e
lim
x→a
ϕ(x)[f(x)−1]
o
.
’
dˆay lim
x→a
ϕ(x)[f(x) −1] du
.
o
.
.
c t´ınh theo c´ac phu
.
o
.
ng ph´ap d˜a nˆeu trˆen

d
ˆa y . N ˆe
´
u lim
x→a
ϕ(x)[f(x) −1] = A th`ı
lim
x→a
[f(x)]
ϕ(x)
= e
A
(57-68).
40 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo
´
57. lim
x→∞


2x +3
2x +1

x+1
(DS. e)
58. lim
x→∞

x
2
− 1
x
2

x
4
(DS. 0)
59. lim
x→0
(1 + tgx)
cotgx
(DS. e)
60. lim
x→0
(1 + 3tg
2
x)
cotg
2
x

(DS. e
3
)
61. lim
x→0

cos x
cos 2x

1
x
2
(DS. e
3
2
)
62. lim
x→
π
2
(sin x)
1
cotgx
(DS. −1)
63. lim
x→
π
2
(tgx)
tg2x

(DS. e
−1
)
64. lim
x→0

tg

π
4
+ x

cotg2x
(DS. e)
65. lim
x→0

cos x

1
x
2
(DS. e
−
1
2
)
66. lim
x→0


cos 3x

1
sin
2
x
(DS. e
−
9
2
)
67. lim
x→0

1+tgx
1 + sin x

1
sin x
(DS. 1)
68. lim
x→
π
4

sin 2x

tg
2
2x

(DS. e
−
1
2
)
Khi t´ınh gi´o
.
iha
.
n c´ac biˆe
’
uth´u
.
cc´och´u
.
a h`am lˆodarit v`a h`am m˜uta
thu
.
`o
.
ng su
.
’
du
.
ng c´ac cˆong th´u
.
c (7.15) v`a (7.16) v`a c´ac phu
.
o

.
ng ph´ap
t´ınh gi´o
.
iha
.
nd˜anˆeuo
.
’
trˆen (69-76).
69. lim
x→e
lnx −1
x − e
(DS. e
−1
)
70. lim
x→10
lgx −1
x −10
(DS.
1
10ln10
)
71. lim
x→0
e
x
2

− 1
√
1 + sin
2
x −1
(DS. 2)
72. lim
x→0
e
x
2
− cos x
sin
2
x
(DS.
3
2
)
7.3. H`am liˆen tu
.
c 41
73. lim
x→0
e
αx
−e
βx
sin αx − sin βx
(DS. 1)

74. lim
x→0
e
sin 5x
−e
sin x
ln(1 + 2x)
(D
S. 2)
75. lim
x→0
a
x
2
− b
x
2
ln cos 2x
, a>0, b>0(DS. −
1
2
ln
a
b
)
76. lim
x→0

a
sin x

+ b
sinx
2

1
x
, a>0, b>0(DS.
√
ab)
7.3 H`am liˆen tu
.
c
D
-
i
.
nh ngh˜ıa 7.3.1. H`am f(x) x´ac di
.
nh trong lˆan cˆa
.
ncu
’
adiˆe
’
m x
0
du
.
o
.

.
cgo
.
i l`a liˆen tu
.
cta
.
id
iˆe
’
md´o n ˆe
´
u
lim
x→x
0
f(x)=f(x
0
).
D
i
.
nh ngh˜ıa 7.3.1 tu
.
o
.
ng du
.
o
.

ng v´o
.
i
D
-
i
.
nh ngh˜ıa 7.3.1
∗
. H`am f(x) x´ac di
.
nh trong lˆan cˆa
.
ncu
’
adiˆe
’
m x
0
du
.
o
.
.
cgo
.
i l`a liˆen tu
.
cta
.

id
iˆe
’
m x
0
nˆe
´
u
∀ε>0 ∃δ>0 ∀x ∈ D
f
: |x −x
0
| <δ⇒|f(x) −f(x
0
)| <ε.
Hiˆe
.
u x − x
0
=∆x du
.
o
.
.
cgo
.
il`asˆo
´
gia cu
’

adˆo
´
isˆo
´
, c`on hiˆe
.
u f(x) −
f(x
0
)=∆f du
.
o
.
.
cgo
.
il`asˆo
´
gia cu
’
a h`am sˆo
´
ta
.
i x
0
tu
.
o
.

ng ´u
.
ng v´o
.
isˆo
´
gia ∆x,t´u
.
cl`a
∆x = x −x
0
, ∆f(x
0
)=f( x
0
+∆x) −f(x
0
).
V´o
.
i ngˆon ng˜u
.
sˆo
´
gia d
i
.
nh ngh˜ıa 7.3.1 c´o da
.
ng

D
-
i
.
nh ngh˜ıa 7.3.1
∗∗
. H`am f(x) x´ac di
.
nh trong lˆan cˆa
.
ncu
’
adiˆe
’
m x
0
du
.
o
.
.
cgo
.
i l`a liˆen tu
.
cta
.
i x
0
nˆe

´
u
lim
∆x→0
∆f =0.
42 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo
´
B˘a
`
ng “ngˆon ng˜u
.
d˜ay” ta c´o d
i
.
nh ngh˜ıa tu
.
o
.

ng d
u
.
o
.
ng
D
-
i
.
nh ngh˜ıa 7.3.1
∗∗∗
. H`am f(x)x´acdi
.
nh trong lˆan cˆa
.
ndiˆe
’
m x
0
∈ D
f
du
.
o
.
.
cgo
.
i l`a liˆen tu

.
cta
.
id
iˆe
’
m x
0
nˆe
´
u
∀(x
n
) ∈ D
f
: x
n
→ x
0
⇒ lim
n→∞
f(x
n
)=f( x
0
).
D
-
i
.

nh l´y 7.3.1. D
iˆe
`
ukiˆe
.
ncˆa
`
nv`adu
’
dˆe
’
h`am f(x) liˆen tu
.
cta
.
idiˆe
’
m
x
0
l`a h`am f( x) tho
’
a m˜ac c´ac diˆe
`
ukiˆe
.
n sau dˆay:
i) H`am pha
’
i x´ac di

.
nh ta
.
imˆo
.
t lˆan cˆa
.
nn`aod´ocu
’
adiˆe
’
m x
0
.
ii) H`am c´o c´ac gi´o
.
iha
.
nmˆo
.
tph´ıa nhu
.
nhau
lim
x→x
0
−0
f(x) = lim
x→x
0

+0
f(x).
iii) lim
x→x
0
−0
= lim
x→x
0
+0
= f(x
0
).
Gia
’
su
.
’
h`am f(x) x´ac di
.
nh trong nu
.
’
a lˆan cˆa
.
n bˆen pha
’
i (bˆen tr´ai)
cu
’

adiˆe
’
m x
0
, ngh˜ıa l`a trˆen nu
.
’
a khoa
’
ng [x
0
,x
0
+ δ) (tu
.
o
.
ng ´u
.
ng: trˆen
(x
0
− δ, x
0
]) n`ao d´o .
H`am f(x)d
u
.
o
.

.
cgo
.
il`aliˆen tu
.
cbˆen pha
’
i(bˆen tr´ai) ta
.
id
iˆe
’
m x
0
nˆe
´
u
f(x
0
+0)=f(x
0
) (tu
.
o
.
ng ´u
.
ng: f(x
0
−0) = f( x

0
)).
D
-
i
.
nh l´y 7.3.2. H`am f(x) liˆen tu
.
cta
.
id
iˆe
’
m x
0
∈ D
f
khi v`a chı
’
khi
n´o liˆen tu
.
cbˆen pha
’
iv`abˆen tr´ai ta
.
idiˆe
’
m x
0

.
H`am liˆen tu
.
cta
.
imˆo
.
tdiˆe
’
m c´o c´ac t´ınh chˆa
´
t sau.
I) Nˆe
´
u c´ac h`am f(x)v`ag(x)liˆen tu
.
cta
.
idiˆe
’
m x
0
th`ı f(x) ±g(x),
f(x) ·g( x)liˆen tu
.
cta
.
i x
0
,v`af(x)/g(x)liˆen tu

.
cta
.
i x
0
nˆe
´
u g(x
0
) =0.
II) Gia
’
su
.
’
h`am y = ϕ(x)liˆen tu
.
cta
.
i x
0
, c`on h`am u = f(y)liˆen
tu
.
cta
.
i y
0
= ϕ(x
0

). Khi d´o h`am ho
.
.
p u = f[ϕ(x)] liˆen tu
.
cta
.
i x
0
.
T`u
.
d´o suy ra r˘a
`
ng
lim
x→x
0
f[ϕ(x)] = f

lim
x→x
0
ϕ(x)

.
H`am f(x)go
.
i l`a gi´an d
oa

.
nta
.
idiˆe
’
m x
0
nˆe
´
u n´o x´ac di
.
nh ta
.
inh˜u
.
ng
d
iˆe
’
mgˆa
`
n x
0
bao nhiˆeu t`uy ´y nhu
.
ng ta
.
ich´ınh x
0
h`am khˆong tho

’
a m˜an
´ıt nhˆa
´
tmˆo
.
t trong c´ac diˆe
`
ukiˆe
.
n liˆen tu
.
co
.
’
trˆen.
7.3. H`am liˆen tu
.
c 43
Diˆe
’
m x
0
du
.
o
.
.
cgo
.

il`a
1) D
iˆe
’
m gi´an doa
.
n khu
.
’
d
u
.
o
.
.
c cu
’
a h`am f(x)nˆe
´
utˆo
`
nta
.
i lim
x→x
0
f(x)=
b nhu
.
ng ho˘a

.
c f(x) khˆong x´ac di
.
nh ta
.
idiˆe
’
m x
0
ho˘a
.
c f(x
0
) = b.Nˆe
´
u
bˆo
’
sung gi´a tri
.
f(x
0
)=b th`ı h`am f(x) tro
.
’
nˆen liˆen tu
.
cta
.
i x

0
,t´u
.
cl`a
gi´an d
oa
.
nc´othˆe
’
khu
.
’
d
u
.
o
.
.
c.
2) Diˆe
’
m gi´an doa
.
nkiˆe
’
uIcu
’
a h`am f(x)nˆe
´
u ∃f(x

0
+0) v`a ∃f(x
0
−0)
nhu
.
ng f( x
0
+0)= f(x
0
− 0).
3) D
iˆe
’
m gi´an doa
.
nkiˆe
’
uIIcu
’
a h`am f(x)nˆe
´
uta
.
idiˆe
’
m x
0
mˆo
.

t trong
c´ac gi´o
.
iha
.
n lim
x→x
0
+0
f(x) ho˘a
.
c lim
x→x
0
−0
f(c) khˆong tˆo
`
nta
.
i.
H`am f(x)d
u
.
o
.
.
cgo
.
il`ah`am so
.

cˆa
´
p nˆe
´
un´od
u
.
o
.
.
cchobo
.
’
imˆo
.
tbiˆe
’
u
th ´u
.
c gia
’
i t´ıch lˆa
.
p nˆen nh`o
.
mˆo
.
tsˆo
´

h˜u
.
uha
.
n ph´ep t´ınh sˆo
´
ho
.
c v`a c´ac
ph´ep ho
.
.
p h`am thu
.
.
chiˆe
.
n trˆen c´ac h`am so
.
cˆa
´
pco
.
ba
’
n.
Mo
.
i h`am so
.

cˆa
´
p x´ac di
.
nh trong lˆan cˆa
.
ncu
’
amˆo
.
tdiˆe
’
m n`ao d´o l `a
liˆen tu
.
cta
.
id
iˆe
’
md´o .
Lu
.
u´yr˘a
`
ng h`am khˆong so
.
cˆa
´
p c´o thˆe

’
c´o gi´an d
oa
.
nta
.
inh˜u
.
ng d
iˆe
’
m
n´o khˆong x´ac di
.
nh c˜ung nhu
.
ta
.
inh˜u
.
ng diˆe
’
m m`a n´o x´ac di
.
nh. D˘a
.
cbiˆe
.
t
l`a nˆe

´
uh`amdu
.
o
.
.
cchobo
.
’
i nhiˆe
`
ubiˆe
’
uth´u
.
c gia
’
i t´ıch kh´ac nhau trˆen c´ac
khoa
’
ng kh´ac nhau th`ı n´o c´o thˆe
’
c´o gi´an doa
.
nta
.
inh˜u
.
ng diˆe
’

m thay dˆo
’
i
biˆe
’
uth´u
.
c gia
’
i t´ıch.
C
´
AC V
´
IDU
.
V´ı du
.
1. Ch´u
.
ng minh r˘a
`
ng h`am f(x) = sin(2x −3) liˆen tu
.
c ∀x ∈ R.
Gia
’
i. Ta lˆa
´
ydiˆe

’
m x
0
∈ R t`uy ´y. X´et hiˆe
.
u
sin(2x − 3) − sin(2x
0
− 3) = 2 cos(x + x
0
− 3) sin(x − x
0
)=α(x).
V`ı |cos(x + x
0
− 3)|  1 v`a sin(x − x
0
)| < |x − x
0
| nˆen khi x → x
0
h`am sin(x −x
0
) l`a h`am vˆo c`ung b´e. T`u
.
d´o suy r˘a
`
ng α(x) l`a t´ıch cu
’
a

h`am bi
.
ch˘a
.
nv´o
.
ivˆoc`ung b´e v`a
lim
x→x
0
sin(2x − 3) = sin(2x
0
−3). 
44 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.
ccu
’
a h`am sˆo
´
V´ı d u
.
2. Ch´u

.
ng minh r˘a
`
ng h`am f( x)=
√
x + 4 liˆen tu
.
cta
.
idiˆe
`
m
x
0
=5.
Gia
’
i. Ta c´o f(5) = 3. Cho tru
.
´o
.
csˆo
´
ε>0. Theo di
.
nh ngh˜ıa 1
∗
ta
lˆa
.

phiˆe
.
u f(x) −f(5) =
√
x +4− 3v`au
.
´o
.
clu
.
o
.
.
ng mˆodun cu
’
a n´o. Ta
c´o
|
√
x +4− 3| =
|x − 5|
|
√
x +4+3|
<
|x −5|
3
(*)
Nˆe
´

u ta cho
.
n δ =3ε th`ı v´o
.
inh˜u
.
ng gi´a tri
.
x m`a |x − 5| <δ=3ε
ta s˜e c´o |
√
x +4−3| <ε.T`u
.
d´o suy r˘a
`
ng h`am f( x)liˆen tu
.
cta
.
idiˆe
’
m
x
0
=5. 
V´ı du
.
3. Ch´u
.
ng minh r˘a

`
ng h`am f(x)=
√
x liˆen tu
.
c bˆen pha
’
ita
.
i
diˆe
’
m x
0
=0.
Gia
’
i. Gia
’
su
.
’
cho tru
.
´o
.
csˆo
´
ε>0t`uy ´y. Bˆa
´

td˘a
’
ng th´u
.
c |
√
x−0| <ε
tu
.
o
.
ng du
.
o
.
ng v´o
.
ibˆa
´
td˘a
’
ng th´u
.
c0 x<ε
2
.Talˆa
´
y δ = ε
2
. Khi d´o

t`u
.
bˆa
´
td˘a
’
ng th ´u
.
c0 x<δsuy r˘a
`
ng
√
x<ε.Diˆe
`
ud´o c´o ngh˜ıa r˘a
`
ng
lim
x→0+0
√
x =0. 
V´ı d u
.
4. Ch´u
.
ng minh r˘a
`
ng h`am y = x
2
liˆen tu

.
c trˆen to`an tru
.
csˆo
´
.
Gia
’
i. Gia
’
su
.
’
x
0
∈ R l`a diˆe
’
mt`uy ´y trˆen tru
.
csˆo
´
v`a ε>0 l`a sˆo
´
cho
tru
.
´o
.
ct`uy ´y. Ta x´et hiˆe
.

u
|x
2
− x
2
0
| = |x + x
0
||x −x
0
|
v`a cˆa
`
nu
.
´o
.
clu
.
o
.
.
ng n´o. V`ı |x + x
0
| khˆong bi
.
ch˘a
.
n trˆen R nˆen dˆe
’

u
.
´o
.
c
lu
.
o
.
.
ng hiˆe
.
u trˆen ta x´et mˆo
.
t lˆan cˆa
.
n n`ao d
´ocu
’
a x
0
,ch˘a
’
ng ha
.
n U(x
0
;1)=
(x
0

− 1; x
0
+ 1). V´o
.
i x ∈U(x
0
; 1) ta c´o
|x + x
0
| = |x −x
0
+2x
0
|  |x −x
0
| +2|x
0
| < 1+2|x
0
|
v`a do d´o
|x
2
− x
2
0
| < (1 + 2|x
0
|)|x −x
0

|.
7.3. H`am liˆen tu
.
c 45
V`ı δ-lˆan cˆa
.
ncu
’
adiˆe
’
m x
0
cˆa
`
n pha
’
in˘a
`
m trong U(x
0
; 1) nˆen ta lˆa
´
y
δ = min

ε
1+2|x
0
|
;1


v`a v´o
.
i |x −x
0
| <δ= min

ε
1+2|x
0
|
;1

ta s˜e
c´o
|x
2
− x
2
0
| <ε. 
V´ı du
.
5. X´ac d
i
.
nh v`a phˆan loa
.
idiˆe
’

m gi´an doa
.
ncu
’
a h`am
f(x)=
1
1+2
1
x−1
·
Gia
’
i. H`am d
˜a cho x´ac di
.
nh ∀x = 1. Nhu
.
vˆa
.
yd
iˆe
’
m gi´an doa
.
nl`a
diˆe
’
m x
0

=1.
Nˆe
´
u(x
n
) l`a d˜ay hˆo
.
itu
.
dˆe
´
n1v`ax
n
> 1th`ı

1
x
n
−1

l`a d˜ay vˆo
c`ung l´o
.
nv´o
.
imo
.
isˆo
´
ha

.
ng dˆe
`
udu
.
o
.
ng. Do d´o

1+2
1
x
n
−1

l`a d˜ay vˆo
c`ung l´o
.
n. T `u
.
d
´o suy r˘a
`
ng f(x
n
)=
1
1+2
1
x

n
−1
l`a d˜ay vˆo c`ung b´e, t´u
.
c
l`a lim
n→∞
f(x
n
) = 0 v`a lim
x→1+0
f(x)=0.
Nˆe
´
u(x
n
) → 1v`ax
n
< 1th`ı

1
x
n
− 1

l`a d˜ay vˆo c`ung l´o
.
nv´o
.
i c´ac

sˆo
´
ha
.
ng d
ˆe
`
u ˆam. Do vˆa
.
y

2
1
x
n
−1

→ 0(n →∞)v`a
f(x
n
)=
1
1+2
1
x
n
−1
→ 1(n →∞),
t´u
.

c l`a lim
x→1−0
f(x) = 1. Do d´odiˆe
’
m x
0
=1l`adiˆe
’
m gi´an doa
.
nkiˆe
’
uI.
V´ı du
.
6. X´ac d
i
.
nh v`a phˆan loa
.
idiˆe
’
m gi´an doa
.
ncu
’
a h`am
f(x)=














x cos
1
x
khi x<0
0 khi x =0
cos
1
x
khi x>0.
46 Chu
.
o
.
ng 7. Gi´o
.
iha
.
n v`a liˆen tu
.

ccu
’
a h`am sˆo
´
Gia
’
i. Diˆe
’
m gi´an doa
.
n c´o thˆe
’
c´o cu
’
a h`am l`a x
0
=0.Tax´et c´ac gi´o
.
i
ha
.
nmˆo
.
tph´ıa ta
.
id
iˆe
’
m x
0

=0.
i) Ta ch´u
.
ng minh r˘a
`
ng lim
x→0−0
f(x) = 0. Thˆa
.
tvˆa
.
y, nˆe
´
u d˜ay (x
n
)
hˆo
.
itu
.
d
ˆe
´
n0v`ax
n
< 0 ∀n th`ı
0  |f(x
n
)| = |x
n

|



cos
1
x
n



 |x
n
|.
V`ı |x
n
|→0 khi n →∞nˆen lim
n→∞
f(x
n
)=0.
ii) H`am d
˜a cho khˆong c´o gi´o
.
iha
.
n bˆen pha
’
ita
.

id
iˆe
’
m x
0
=0. Dˆe
’
ch´u
.
ng minh diˆe
`
ud´o t a x ´et hai d˜ay hˆo
.
itu
.
dˆe
´
n0lˆa
.
pnˆent`u
.
c´ac d˜ay
sˆo
´
du
.
o
.
ng x
n

=
1
π
2
+ nπ
v`a x

n
=
1
2πn
.Nˆe
´
unhu
.
h`am f c´o gi´o
.
iha
.
n
bˆen pha
’
ita
.
idiˆe
’
m x
0
= 0 th`ı hai d˜ay f(x
n

)v`af(x

n
) pha
’
ihˆo
.
itu
.
dˆe
´
n
c`ung mˆo
.
t gi´o
.
iha
.
n. Thˆe
´
nhu
.
ng f(x

n
)=cos2πn =1hˆo
.
itu
.
dˆe

´
n 1, c`on
f(x
n
) = cos

π
2
+ nπ

=0hˆo
.
itu
.
dˆe
´
n0.
T`u
.
d´o suy r˘a
`
ng h`am c´o gi´an doa
.
nkiˆe
’
uIIta
.
idiˆe
’
m x

0
=0. 
V´ı d u
.
7. T`ım v`a phˆan loa
.
ic´acd
iˆe
’
m gi´an doa
.
ncu
’
a c´ac h`am:
1) y = (signx)
2
;2)y =[x]
Gia
’
i
1) T`u
.
di
.
nh ngh˜ıa h`am signx suy r˘a
`
ng
(signx)
2
=




1,x=0
0,x=0.
T`u
.
d
´o suy r˘a
`
ng h`am y = (signx)
2
liˆen tu
.
c ∀x = 0 (h˜ay du
.
.
ng d
ˆo
`
thi
.
cu
’
a h`am) v`a ta
.
id
iˆe
’
m x

0
= 0 ta c´o y(0 − 0) = y(0 + 0) = y(0).
Diˆe
`
ud´o c´o ngh˜ıa r˘a
`
ng x
0
=0l`adiˆe
’
m gi´an doa
.
nkhu
.
’
du
.
o
.
.
c.
2) Gia
’
su
.
’
n ∈ Z.Nˆe
´
u n − 1  x<nth`ı [x]=n − 1, nˆe
´

u
n  x<n+1 th`ı [x]=n (h˜ay du
.
.
ng d
ˆo
`
thi
.
cu
’
a h`am phˆa
`
n nguyˆen
[x]). Nˆe
´
u x
0
∈ Z th`ı tˆo
`
nta
.
i lˆan cˆa
.
ncu
’
adiˆe
’
m x
0

(khˆong ch´u
.
a c´ac sˆo
´