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

.
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
`

’
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
.

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

|
√
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
⇔


.
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

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
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

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

− 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

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

´
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

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

).
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(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

.
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

’
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

; 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
+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

´
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
.
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

)
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

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
.

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

.
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

)
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
.

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

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
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 −

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→∞

´
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
’

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

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

.
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

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 −

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

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→∞


(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

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

.
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

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

.
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
.

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
’

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

´
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
.

.
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 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

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
.

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

.
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

.
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

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

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

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
.

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
.

’
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

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

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

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
´

`
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
´

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

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
’

− 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


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
’

=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

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

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)=



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

|



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
.

=
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

´
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




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

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