Tài liệu Giáo trình giải tích docx

TRƯỜNG ĐẠI HỌC ĐÀ LẠT
KHOA TOÁN - TIN HỌC
Y  Z



TẠ LÊ LI






GIẢI TÍCH 1

(Giáo Trình)

Lưu hành nội bộ
Y Đà Lạt 2008 Z
R
q
q
2
=2
√
2
R
R
• R
+:R × R → R, (x, y) → x + y
· : R × R → R, (x, y) → xy
∀x, y x + y = y + x
∀x, y, z (x + y)+z = x +(y + z)
∃0, ∀x, x +0 = x (0 )
∀x, ∃−xx+(−x)=0 (−x x)
∀x, y xy = yx
∀x, y, z (xy)z = x(yz)
∃1 =0, ∀x 1x = x (1 )
∀x =0, ∃x
−1
xx
−1
=1 (x
−1

x)
∀x, y, z x(y + z)=xy + xz
• R ≤
∀x, y x ≤ y y ≤ x
∀xx≤ x
∀x, y x ≤ y, y ≤ x ⇒ x = y
∀x, y, z x ≤ y, y ≤ z ⇒ x ≤ z
∀x, y, z x ≤ y ⇒ x + z ≤ y + z
∀x, y 0 ≤ x, 0 ≤ y ⇒ 0 ≤ xy
• R
R
R
R

R R

n

i=1
x
i
= x
1
+ ···+ x
n
n

i=1
x
i

= x
1
···x
n
x − y = x +(−y)
x
y
= xy
−1
x ≤ y y ≥ x x y y x
x<y y>x x ≤ y x = y x y y x
0 <x x x<0 x
(a, b)={x ∈ R : a<x<b}
[a, b]={x ∈ R : a ≤ x ≤ b}
[a, b), (a, b]
R
O 0 1 =0 1
0 1 M
OM M 1 0
✲
0
t
1 M>0M

< 0
A ⊂ R
b ∈ R x ≤ b, ∀x ∈ A
b
A
A ⊂ R

a ∈ R a ≤ x, ∀x ∈ A
a
A
b
∗
A b
∗
=supA b
∗
A
a
∗
A a
∗
=infA a
∗
A
A = {
1
2
,
3
4
, ··· ,
2
n
−1
2
n
, ···} sup A =1, inf A =

1
2
A = {q : q q
2
< 2}
a
∗
=infA b
∗
=supA R A
a
∗
b
∗
q q
2
=2
R
sup A ∈ A inf A ∈ A A
M
A M =maxA M =supA M ∈ A
m
A m = min A m =infA m ∈ A
A ⊂ R a =supA
a A ∀>0, ∃x

∈ A : a − <x

N, Z, Q
N = {n : n =0 n =

n
  
1+···+1}
Z = {p : p ∈ N − p ∈ N }
Q = {
p
q
: p ∈ Z,q∈ N,q=0}/ ∼,
p

q

∼
p
q
⇔ pq

− qp

=0
x ∈ R
x
|x| =

x x ≥ 0
−x x<0
x, y
|x|≥0, |xy| = |x||y|, |x + y|≤|x|+ |y|
x ∈ R n ∈ N x<n
n ≤ x, ∀n ∈ N

a =supN
sup n
0
∈ N a−1 <n
0
a<n
0
+1 ∈ N 
x, y > 0 n ∈ N x<ny
x>0 n ∈ N 0 <
1
n
<x
x>0 n ∈ N n ≤ x<n+1
x ∈ R
[x]= n n ≤ x<n+1
[0, 5], [−2, 5], [0, 0001]
R
x, y ∈ R x<y r ∈ Q x<r<y
x ∈ R >0 r ∈ Q |x − r| <
n ∈ N 0 <
1
n
<y−x
m ∈ N m ≤ nx < m +1
m
n
≤ x<
m +1
n

r =
m +1
n
∈ Q x<r=
m +1
n
=
m
n
+
1
n
<x+(y − x)=y 
R
n x>0 n ∈ N \{0}
y>0 y
n
= x
y
n x y =
n
√
x
A = {t ∈ R : t
n
≤ x} A = ∅ t =0
1+x y =supA
y
n
= x

y
n
<x 0 <h<1
(y + h)
n
≤ y
n
+ h(
n

k=1
C
k
n
y
n−k
)=y
n
+ h((y +1)
n
− y
n
)
0 <h<
x − y
n
(y +1)
n
− y
n

h<1 (y + h)
n
<x y + h ∈ A
y + h>y=supA
y
n
>x k>0 (y −k)
n
>x y −k
A y =supA 
R
√
2,
√
3,
3
√
5,
4
√
16
R
∞
R = R ∪{+∞, −∞}
x ∈ R −∞ <x<+∞
x +(+∞)=+∞,x+(−∞)=−∞
x(+∞)=+∞ x>0,x(+∞)=−∞ x<0
x
+∞
=

x
−∞
=0
∞−∞, 0 ∞,
∞
∞
A inf A = −∞ sup A =+∞
1 3
00, 30, 33 0, 333 0, 3333 ···
1
1
2
1
2
2
1
2
3
1
2
4
···
122
2
2
3
2
4
···
010101 ···

•
•
1
3
0
X ⊂ R X
(x
n
)
n∈N
= x
0
,x
1
,x
2
,x
3
, ···
X x : N → X, n → x
n
= x(n)
R
2
{ (n, x
n
): n ∈ N }
0
s
1

s
2
s
3
s
q
s
q
s
n
sx
q
s
q
s
q
s
s
s
s
s
✲
+∞
✻
x
N = {0, 1, 2, ···} n ∈ N n +1∈ N
0 < 1 < 2 < 3 < ···
• Σ={0, 1, ··· ,N}
(x
0

,x
1
,x
2
, ···) x
n
∈ Σ
• x
n
=3.10
−1
+3.10
−2
+ ···+3.10
−n
,
x
n
=
1
2
n
,x
n
=2
n
x
n
=1−(−1)
n

• x
n
= n! x
0
=1,x
n+1
=(n +1)x
n
(n ≥ 1)
x
0
∈ R x
n+1
= f(x
n
) n =0, 1, ··· f
x
0
=0,x
1
=1,x
n+1
= x
n
+ x
n−1
(n ≥ 2)
f(x)=
√
1+x f(x)=4λx(1 − x) λ ∈{0.7, 0.8, 0.9}

x
n+1
= f(x
n
) x
0
=1
x
n
=
n
{x
n
: n ∈ N} (x
n
)
n∈N
a ∈ R
(x
n
)
n∈N
>0
N

 n>N

|x
n
− a| <

∀>0, ∃N : n>N ⇒|x
n
− a| <
(x
n
) a
lim
n→∞
x
n
= a lim x
n
= a x
n
→ a, n →∞
0
s
1
s
2
s
3
s
q
s
q
s
N
s
n

s
q
s
q
s
q
s
q
s
q
s
q
s
a + 
a − 
a
✲
+∞
✻
x
•
• lim
n→∞
x
n
= a lim
n→∞
|x
n
− a| =0

•
{(x, y): y = a } R
2
• (x
n
) a b
(x
n
) |a −b|≤|a −x
n
|+|x
n
−b|→0 n →∞ |a −b| =0 a = b
x
n
=
1
√
n
n =1, 2, ···
lim
n→∞
x
n
=0

1
1
10
1

100
1
1.000
1
1.000.000
N

1 100
 N

0 <
1
<
2
⇒ N

1
≥ N

2
lim
n→∞
x
n
= a |x
n
− a|
|x
n
− a|≤f(N) n>N N

 f(N) <
>0 N n>N |x
n
−a|≤f(N) <
lim
n→∞
1
n
p
=0 p>0
n>N |
1
n
p
− 0| =
1
n
p
<
1
N
p
>0 N>
p

1

N =[
p


1

]+1
n>N |
1
n
p
− 0| <
1
N
p
<
(x
n
)=0 0, 30, 33 0, 333 0, 3333 ···→
1
3
>0 N =[3/] n>N
|x
n
−
1
3
| = |0, 33 ···3
  
n
−
1
3
| <

3
10
n
<
3
10
N
<
3
N
<
•
(2
n
)
lim
n→∞
x
n
=+∞ ∀E>0, ∃N : n>N⇒ x
n
>E
lim
n→∞
x
n
= −∞ ∀E>0, ∃N : n>N⇒ x
n
< −E
•

0 1
lim
n→+∞
a
n
=









0 |a| < 1
1 a =1
+∞ a>1
a ≤−1
(x
n
)
n
0
<n
1
< ···<n
k
< ··· (x
n

k
)
k∈N
(x
n
)
(x
n
)
N −→ N −→ R
k → n(k)=n
k
→ x
n
k
= x
n(k)
a ∈ R
a ((−1)
n
)
(1) (−1)
1 −1
• (x
n
) a a
• a (x
n
) >0
n ∈ N |x

n
− a| <
lim sup
n→∞
x
n
= lim
n→∞
x
n
=sup{a : a (x
n
)}
lim inf
n→∞
x
n
= lim
n→∞
x
n
=inf{a : a (x
n
)}
x
n
=(−1)
n
lim sup x
n

=1 lim inf x
n
= −1
x
n
=(−1)
n
n lim sup x
n
=+∞ lim inf x
n
= −∞
x
n
=sin
nπ
2
lim sup x
n
= lim inf x
n
=
x
0
∈ R n ≥ 1
x
n
=

3x

n−1
+1 x
n−1
1
2
x
n−1
x
n−1
x
0
=17 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ···
1 1, 4, 2, 1, 4, 2, 1, ···
x
0
n x
n
=1
• lim sup x
n
lim inf x
n
∞
• lim inf x
n
≤ lim sup x
n
• (x
n
) lim inf x

n
= lim sup x
n
• lim sup x
n
= M >0 x
n
>M−
x
n
>M+ 
• lim inf x
n
= m >0 x
n
<m+ 
x
n
<m−
(x
n
) M |x
n
| <M,∀n
(x
n
) (y
n
)
(x

n
+ y
n
) (x
n
y
n
)

x
n
y
n

lim
n→∞
y
n
=0
lim
n→∞
(x
n
+y
n
) = lim
n→∞
x
n
+ lim

n→∞
y
n
, lim
n→∞
(x
n
y
n
) = lim
n→∞
x
n
lim
n→∞
y
n
, lim
n→∞
x
n
y
n
=
lim
n→∞
x
n
lim
n→∞

y
n
(x
n
) (y
n
) n
x
n
≤ y
n
lim
n→∞
x
n
≤ lim
n→∞
y
n
n x
n
≤ y
n
≤ z
n
lim
n→∞
x
n
=

lim
n→∞
z
n
= a lim
n→∞
y
n
= a
lim
n→∞
x
n
= a lim
n→∞
y
n
= b
 =1 N |x
n
− a| < 1, ∀n>N
M =max{|x
0
|, ··· , |x
N
|, |a| +1} |x
n
| <M,∀n
|(x
n

+ y
n
) − (a + b)|≤|x
n
− a| + |y
n
− b|
|x
n
y
n
− ab|≤|x
n
y
n
− x
n
b + x
n
b − ab|≤M|y
n
− b| + |b||x
n
− a|
b =0  = |b| /2 N |y
n
− b| < |b|/2 ∀n>N
n>N |y
n
| = |b − b + y

n
|≥|b|−|y
n
− b| > |b|/2




x
n
y
n
−
a
b




=




x
n
b − y
n
a
by

n




=




x
n
b − ab
by
n
+
ab − y
n
a
by
n




≤
|x
n
− a|
|y

n
|
+
|a||b −y
n
|
|by
n
|
≤
|x
n
− a|
|b|/2
+
|a||b −y
n
|
|b||b|/2
n → +∞ → 0
n x
n
≤ y
n
a>b
 =
a−b
2
> 0 n |x
n

− a| < |y
n
− b| <
y
n
<b+  =
a+b
2
= a − <x
n
>0 lim x
n
= lim z
n
= a N
1
|x
n
− a| <,|z
n
− a| < n>N
1
N
2
x
n
≤ y
n
≤ z
n

, ∀n ≥ N
2
n ≥ max(N
1
,N
2
)
−<x
n
− a ≤ y
n
− a ≤ z
n
− a< |y
n
− a| <
lim y
n
= a 
• ((−1)
n
)
• (x
n
), (y
n
) x
n
<y
n

, ∀n lim
n→∞
x
n
≤ lim
n→∞
y
n
lim
n→∞
x
n
= a lim
n→∞
|x
n
| = |a| lim
n→∞
p

|x
n
| =
p

|a|
lim
n→∞
n
2

− 3n +6
3n
2
+4n +2
lim
n→∞
√
n(
√
n +2−
√
n +1)
n
2
n
n
2
lim
n→∞
n
2
− 3n +6
3n
2
+4n +2
= lim
n→∞
n
2
(1 − 3/n +6/n

2
)
n
2
(3 + 4/n +2/n
2
)
= lim
n→∞
1 − 3/n +6/n
2
3+4/n +2/n
2
=
1 − lim 3/n + lim 6/n
2
3 + lim 4/n + lim 2/n
2
=
1 − 0+0
3+0+0
=
1
3
lim
n→∞
√
n(
√
n +2 −

√
n + 1) = lim
n→∞
√
n
(
√
n +2−
√
n +1)(
√
n +2+
√
n +1)
√
n +2+
√
n +1
= lim
n→∞
√
n
(n +2)− (n +1)
√
n +2+
√
n +1
= lim
n→∞
√

n
√
n(

1+
2
n
+

1+
1
n
)
= lim
n→∞
1

1+
2
n
+

1+
1
n
=
1
(lim

1+

2
n
+ lim

1+
1
n
)
=
1
√
1+
√
1
=
1
2
Q Q x
n
=(1+
1
n
)
n
R
(x
n
≤ x
n+1
, ∀n)&(∃M,x

n
<M,∀n) ⇒∃lim x
n
(x
n
≥ x
n+1
, ∀n)&(∃m, m < x
n
, ∀n) ⇒∃lim x
n
(x
n
)
(−x
n
) (x
n
)
a =sup{x
n
: n ∈ N}
lim x
n
= a >0
x
n
≤ a x
N
a −<x

N
n>N a − <x
n
≤ a<a+  |x
n
− a| <
lim x
n
= a 
(x
n
) lim x
n
=+∞
(x
n
) lim x
n
= −∞
I
n
=[a
n
,b
n
]
I
n
⊃ I
n+1

n ∈ N I
n
∩
n∈N
I
n
= ∅
a
n
≤ a
n+1
≤ b
n+1
≤ b
n
(a
n
)
(b
n
)
a = lim a
n
lim b
n
= b a ≤ b
[a, b] ⊂ I
n
, ∀n 
a

0
≤ x
n
≤ b
0
, ∀n I
0
=[a
0
,b
0
]
x
n
I
1
n
1
x
n
1
∈ I
1
I
1
x
n
I
2
n

2
>n
1
x
n
2
∈ I
2
I
0
⊃ I
1
⊃···⊃I
k
I
k
b
0
−a
0
2
k
n
1
<n
2
< ···<n
k
x
n

k
∈ I
k
a ∈ I
k
, ∀k |x
n
k
−a|≤
b
0
−a
0
2
k
→ 0
k →∞ (x
n
k
)
k∈N
a 
(x
n
) (x
n
)
∀>0, ∃N : n, m > N ⇒|x
n
− x

m
| <
⇐ limx
n
= a >0 N |x
n
− a| </2
∀n>N m, n > N |x
n
− x
m
|≤|x
n
− a| + |x
m
− a| </2+/2=
⇒ (x
n
)
(x
n
)  =1 N x
N
−1 <x
n
<x
N
+1, ∀n>N
M =max{|x
0

|, ··· , |x
N
|, |x
N
| +1} |x
n
|≤M,∀n
(x
n
k
)
k∈N
a
(x
n
) a |x
k
−a|≤|x
k
−x
n
k
|+|x
n
k
−a|
n
k
≥ k k →∞ n
k

→∞ |x
k
− x
n
k
|→0
|x
n
k
− a|→0 a lim
k→∞
x
k
= a 
|x
n
− x
n+p
|→0 , n →∞, p =0, 1, ···
lim
n→∞
1
n
p
=0 (p>0)
lim
n→∞
n
√
a =1 (a>0)

lim
n→∞
n
√
n =1
lim
n→∞
n
√
n!=+∞
lim
n→∞
n
p
a
n
=0 (a>1)
lim
n→∞
a
n
=0 |a| < 1 lim
n→∞
a
n
=+∞ a>1
a ≥ 1 x
n
=
n

√
a − 1 lim x
n
=0
x
n
≥ 0 a =(1+x
n
)
n
≥ 1+nx
n
0 ≤ x
n
≤
a − 1
n
lim x
n
=0
0 <a<1
1
a
x
n
=
n
√
n − 1
n =(1+x

n
)
n
≥
n(n − 1)
2
x
2
n
0 ≤ x
n
≤
√
2
√
n − 1
lim x
n
=0 lim
n
√
n =1
n! >

n
3

n
n
√

n! >
n
3
a>1 a
1
p
=1+u (u>0)
(a
1
p
)
n
=(1+u)
n
>
n(n − 1)
2
u
2
lim
n
p
a
n
= lim


n
(a
1

p
)
n


p
=0
p =0 
s
n
=1+
1
1!
+
1
2!
+ ···+
1
n!
t
n
=

1+
1
n

n
lim
n→∞

s
n
= lim
n→∞
t
n
= e
(s
n
) s
n
=1+1+
1
1.2
+
1
1.2.3
+ ··· +
1
1.2 n
< 1+
1+
1
2
+
1
2
2
+ ···+
1

2
n−1
< 3 lim s
n
= e
t
n
=

1+
1
n

n
=
n

k=0
n!
k!(n − k)!
1
n
k
=
n

k=0
1
k!
n

n
n − 1
n

n − k +1
n
=
n

k=0
1
k!

1 −
1
n



1 −
k −1
n

t
n
<t
n+1
t
n
≤ s

n
< 3 lim t
n
= e

e = e

t
n
≤ s
n
e

≤ e
n ≥ m
t
n
=1+1+
1
2!

1 −
1
n

+ ···+
1
n!

1 −

1
n



1 −
n − 1
n

≥ 1+1+
1
2!

1 −
1
n

+ ···+
1
m!

1 −
1
n



1 −
m − 1
n


m n →∞ e

≥ 1+1+
1
2!
+ ···+
1
m!
= s
m
m →∞ e

≥ e 
e e =2, 71828 ···
e =
m
n
∈ Q
0 <e−s
n
=
1
(n +1)!
+ ···<
1
n!n
0 <n!(e −s
n
) <

1
n
n!e, n!s
n

x
n
= a
0
+ a
1
x + ···+ a
n
x
n
|x| < 1 |a
k
| <M,∀k
x
n
=1+
1
2
+ ···+
1
n
|x
n+p
− x
n

| = |a
n+1
x
n+1
+ ···+ a
n+p
x
n+p
|≤|a
n+1
|x|
n+1
| + ···+ |a
n+p
||x
n+p
|
≤ M|x|
n+1
+ ···+ M|x|
n+p
≤ M|x|
n+1
(1 + ···+ |x|
p
)
≤ M|x|
n+1
1
1 −|x|

n →∞ |x
n+p
− x
n
|→0 p (x
n
)
n, m =2n |x
m
− x
n
| =
1
n +1
+ ···+
1
2n
>
1
2
(x
n
)

x ∈ R
a
0
=[x] ∈ Z,a
n
=[10

n
(x − a
0
−
a
1
10
−···−
a
n−1
10
n−1
)] ∈{0, 1, ··· , 9},
x
n
= a
0
+
a
1
10
+ ···+
a
n
10
n
→ x, n →∞
x = a
0
,a

1
a
2
···a
n
···
R
a
0
=[x] a
0
≤ x<a
0
+1 0 ≤ x − a
0
< 1
a
1
= [10(x − a
0
)] ∈{0, 1, ··· , 9}
a
1
10
≤ x − a
0
<
a
1
+1

10
[0, 1] x −a
0
0 ≤ x − a
0
−
a
1
10
<
1
10
a
2
∈{0, 1, ··· , 9}
a
2
10
2
≤ x − a
0
−
a
1
10
<
a
2
+1
10

2
n 0 ≤ x − a
0
−
a
1
10
−···−
a
n
10
n
<
1
10
n
a
n+1
=[10
n+1
(x − a
0
−
a
1
10
−···−
a
n
10

n
)] a
n+1
∈{0, 1, ··· , 9}
0 ≤ x − a
0
−
a
1
10
−···−
a
n
10
n
−
a
n+1
10
n+1
<
1
10
n+1
x
n
0 ≤ x − x
n
<
1

10
n
lim x
n
= x 
•
1, 000 ···=0, 999 ··· 0, 5=0, 4999 ···
•
1
2
=0, 5 ,
1
3
=0, 333 ··· , 0, 123123123 ···= 123 ×
1
10
3
− 1
R
X, Y
X Y
n {1, 2, ··· ,n}
N
N → X X
2N, Z, Q
R
a, b ∈ R a = b [a, b]
[a, b]={x
n
: n ∈ N} [a, b]

I
1
x
1
∈ I
1
I
1
I
2
x
2
∈ I
2
I
1
⊃ I
2
⊃···⊃I
n
⊃··· x
n
∈ I
n
x ∈∩
n∈N
I
n
x ∈ [a, b]
x = x

n
, ∀n x ∈ [a, b] 
• N
X ⊂ N N X
0 → x
0
= min X, n → min(X \{x
0
, ··· ,x
n−1
})
• X f : X → Y Y
m : Y → X, m(y) = min f
−1
(y) m
Y → m(Y )
• N
2
f : N
2
→ N,f(m, n)=
(m + n)(m + n +1)
2
+ n )
✲
N
✻
N
r
r

r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r❜❜❜❜❜❜
❜❜❜❜❜❜
❜❜❜❜❜❜
❜❜❜❜❜❜
❜❜❜❜❜❜
❜❜❜❜❜❜
❅
❅
❅
❅
❅
❅

❅
❅
❅
❅
❅■
❅
❅
❅
❅
❅
❅
❅
❅
❅■
❅
❅
❅
❅
❅
❅
❅■
❅
❅
❅
❅
❅
❅■
❅
❅
❅

❅■
• (X
n
)
n∈I
X = ∪
n∈I
X
n
N → I n → i
n
n N → X
n
,m→ f
n
(m)
N
2
→ X, (m, n) → f
i
n
(m)
• 0 1
X
N → X, n → x
n
x
0
= x
0,0

x
0,1
x
0,2
··· ···
x
1
= x
1,0
x
1,1
x
1,2
··· ···
x
2
= x
2,0
x
2,1
x
2,2
··· ···
x
n
= x
n,0
x
n,1
x

n,2
··· x
n,n
···
y =(y
n
) y
n
=1
x
n,n
=0 y
n
=0 x
n,n
=1 y X 0, 1
X y = x
n
, ∀n
n!
n!=

n
e

n
√
2πne
θ
n

12n
, 0 <θ
n
< 1

f : X → Y, x → y = f(x)
X, Y R
x ∈ X
y = f(x) ∈ Y
X
f
f(X)={y ∈ R : ∃x ∈ X, y = f(x)}
f
y x
y =2πx, y = mx, y = mx
2
{x ∈ R : f(x) }
f(x)=
√
x − 1
x − 2
{x ∈ R : x − 1 ≥ 0,x− 2 =0} =[1, 2) ∪(2, +∞)
f(x)=[x]=n n ≤ x<n+1
f(x)= x =





−1 x<0

0 x =0
+1 x>0
D χ
D
(x)=

1 x ∈ D
0 x ∈ D
[1, 5], [−π], [e], [sin x], (−2), (2
64
), (−[0, 3])
f = {(x, y):x ∈ X, y = f(x)} R × R = R
2
R
2
(0, 0) O R ×0
Ox 0 ×R Oy (x, y) ∈ R
2
Ox (x, 0)
Oy (0,y)
f
f
s
O
✲
x
✻
y
s
(x, f (x))

(x
0
,f(x
0
)), (x
1
,f(x
1
)), ··· , (x
n
,f(x
n
))
f
[x] (x)
x
x
0
x
1
··· x
n
y y
0
y
1
··· y
n
f,g : X → R
f ± g, fg,

f
g
g(x) =0, ∀x ∈ X
(f ±g)(x)=f(x) ±g(x),fg(x)=f(x)g(x),
f
g
(x)=
f(x)
g(x)
,x∈ X
f : X → Y g : Y → Z g ◦f : X → Z
g ◦f(x)=g(f(x))
f : X → Y f
−1
: Y → X
f
−1
(y)=x ⇔ y = f(x)
f(x)=x − [x]
f(x)=[x] g(x)= (x) f ◦ g g ◦ f
f X
∀x
1
,x
2
∈ X, x
1
<x
2
⇒ f(x

1
) ≤ f(x
2
)( f(x
1
) <f(x
2
))
f
X
∀x
1
,x
2
∈ X, x
1
<x
2
⇒ f(x
1
) ≥ f(x
2
)( f(x
1
) >f(x
2
))
f(x)=x
n
n ∈ N [0, +∞)

f(x)=[x] g(x)= (x) R
n f(x)=x
n
R
X x ∈ X −x ∈ X
f
X f(−x)=f(x), ∀x ∈ X
f
X f(−x)=−f(x), ∀x ∈ X
x
2
, cos x x
3
, sin x R
f
f(x)=
1
2
(f(x)+f(−x)) +
1
2
(f(x) − f(−x))
Oy
O (x, y = f(x))
✲
x
✻
y
0
s

(x, y)
s
(y, x)
s
(−x, y)
s
(−x, −y)
s
(x + T,y)












y = x
f X
T>0
f(x + T )=f(x), ∀x ∈ X
T
f
x ∈ X x + T ∈ X x + nT ∈ X n ∈ N
f(x + nT )=f(x)
T

k ∈ Z \{0} sin kx cos kx
2π
k
f(x)=x − [x] 1
Q χ
Q
•
x
α
e
x
ln x sin x arctan x
x
exp(x)=e
x
= lim
n→+∞

1+
x
n

n
R (0, +∞)
e
0
=1,e
x+x

= e

x
e
x






1+
t
n

n
− 1




≤|t|(e − 1) |t|≤1(∗)

1+
t
n

n
=1+t
n

k=1

C
k
n
t
k−1
n
k
|t|≤1





1+
t
n

n
− 1




≤|t|
n

k=1
C
k
n

|t|
k−1
n
k
≤|t|
n

k=1
C
k
n
|t|
k−1
n
k
≤|t|
n

k=1
C
k
n
1
n
k
= |t|

1+
1
n


n
− 1

≤|t|(e − 1)
x ∈ R x
n
=

1+
x
n

n
(x
n
)
x>0 e (x
n
)
N ∈ N x ≤ N
x
n
≤

1+
N
n

n

≤

1+
1
n

N.n
≤

1+
1
n

n.N
≤ 3
N
exp(x) = lim
n→+∞
x
n
x ≥ 0
x<0 −x>0

1+
x
n

n
=
(1 −

x
2
n
2
)
n
(1 −
x
n
)
n
(∗) t =
x
2
n
lim
n→∞
(1 −
x
2
n
2
)
n
=1
lim
n→+∞

1+
x

n

n
=
1
exp(−x)
exp(x) x ∈ R
e
0
=1

1+
x
n

n

1+
x

n

n

1+
x+x

n

n

=

1+
xx

n
2
(1 +
x+x

n
)

n
n →∞ (∗) t =
xx

n(1 +
x+x

n
)
e
x
e
x

e
x+x


=1
e
x
> 0 e
t
> 1 t>0 x<x

e
x
− e
x

= e
x
(1 − e
x

−x
) < 0
e
x

ln x e
x
(0, +∞) R
ln e =1, ln x +lnx

=lnxx

x

α
α ∈ R
n ∈ N x
n
= x ···x n
R n n (−∞, 0)
[0, +∞)
n ∈ N x
−n
=
1
x
n
R \ 0 n n
(−∞, 0) (0, +∞)
✲
✻
y = x
2n
✲
✻
y = x
2n+1
✲
✻
y =
1
x
2n+1
✲

✻
y =
1
x
2n
n ∈ N
n
√
x = x
1
n
x
n
n
R n [0, +∞)
✲
✻
y =
2n
√
x
✲
✻
y =
2n+1
√
x
m, n ∈ Z,n>0 x
m
n

=(
n
√
x)
m
n m
α x
α
= e
α lnx
(0, +∞) α>0 α<0
(xx

)
α
= x
α
x

α
a
x
= e
x ln a
a>0
R (0, +∞) a>1
0 <a<1
a
x+x


= a
x
a
x

✲
x
✻
y
r
1
y = a
x
(a>1)
✲
x
✻
y
r
1
y = a
x
(0 <a<1)
log
a
x =
ln x
ln a
(a>0,a=1
(0, +∞) R a>1

0 <a<1
log
a
x +log
a
x

=log
a
xx

log
a
x =log
a
b log
b
x
log
a
x
α
= α log
a
x
a
x
log
a
x y =log

a
x ⇔ a
y
= x