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+

=
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
≤ 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

= 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

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

<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

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

=
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



n
(a
1
p
)
n


p
=0
p =0 
s
n
=1+
1
1!
+
1
2!
+ ···+
1
n!
t
n
=

1+
1

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

<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 −
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)!
+ ···<

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

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

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

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

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
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

= 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






−1 x<0
0 x =0
+1 x>0
D χ
D
(x)=

1 x ∈ D
0 x ∈ D
[1, 5], [−π], [e], [sinx], (−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
✲

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

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))
✲

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

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

= |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
≤


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

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

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
✲
✻