dukazy mati kopec
TRANSCRIPT
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ě ů řě
:
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........................................................................................................................................................ 8
*V ................................................................................................................................. 8
M ůů .................................................................................................................................... 8
*Př ů ....................................................................................................................................... 8
*Nř ů ................................................................................................................................... 8
*Dů .................................................................................................................................... 8
*M ........................................................................................................................ 8
M ................................................................................................................................................... 8
D : ............................................................................................................................ 8
D , : ................................................................................................ 8
Vě 1 M ..................................................................................................... 8
Vě 2 C ........................................................................................................ 9
Č ................................................................................................................................. 10
M ř č ............................................................................................................. 10
M č ...................................................................................................................... 10
M č ............................................................................................................ 10
R č .................................................................................................................................... 11
V č : .................................................................. 11
V ř č : ............................................................................ 11
A : ................................................................................................................................ 11
K č .............................................................................................................................. 12
Vě ě .......................................................................................................... 12
D ........................................................................................................................... 12
Vě 3 ....................................................................................................................... 12
Vě 4 č .......................................................................................................... 13
Vě 5 A .................................................................................................... 14
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Vě 6 ě ............................................................................................................ 14
Vě 7 ě ℚ ℝ ∖ ℚ ........................................................................................................ 16L ............................................................................................................................... 17
.................................................................................................................................................. 17
D .................................................................................................................... 17
D .................................................................................................... 17
D ...................................................................................................... 18
D ........................................................................................................................ 18
K ................................................................................................................ 18
D .............................................................................................. 18
Vě 8 č ....................................................................................................... 18
Vě 9 . .............................................................. 19
D ...................................................................................................... 20
Vě 10 .......................................................................................... 20
Vě 11 .............................................................................................................. 20
Vě 12 ..................................................................................................... 22
Vě 13 ř ........................................................................................................ 22
Vě 14 ......................................................................................................... 24
N ............................................................................................................ 24
D .......................................................................................... 24
Vě 9 ............................................................................. 25
Vě 14 ..................................................................................................... 25
D ř ........................................................................................................ 25
Vě 11 .............................................................................................. 26
Vě 15 ě ..................................................................................................... 26
Vě 16 ě .................................................................................. 26
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Vě 17 BW ..................................................................................................... 26
Z ............................................................................................................................................... 28
.................................................................................................................................................. 28
D : ........................................................................................................................ 28
Zč: ......................................................................................................................................... 28
*Vč ...................................................................................................................................... 29
*P ........................................................................................................................................... 29
Dč ................................................................................................................ 29
D ................................................................................................................................ 29
D .................................................................................................................. 29
*Z ......................................................................................................................... 29
*S ........................................................................................................................ 29
*R ............................................................................................................. 29
*P .......................................................................................................................... 29
*Z .......................................................................................... 29*B ......................................................................................................................................... 29
*I .............................................................................................. 29
*I ........................................................................................................................................ 29
F ě ............................................................................................................ 29
.................................................................................................................................................. 29
D .............................................................................................................................. 29
D / .................................................................................................. 30
D ........................................................................................................... 30
D ........................................................................................................................ 30
L ..................................................................................................................................... 30
D ........................................................................................................................ 30
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D ě ........................................................................................................ 30
Vě 18 č ......................................................................................... 31
*D ......................................................................................... 31
*D ...................................................................................................... 31
D .............................................................................................................. 31
Vě 19 ........................................................................................................ 31
*Vě 20 ............................................................................................................ 32
Vě 21 ě ..................................................................................................... 35
Vě 22 , 2 .......................................................... 35
Vě 23 ..................................................................................................... 38
Vě 24 H () ............................................................................................................ 39
D .......................................................................................................................... 40
Vě 25 ............................................................................................... 40
F ............................................................................................................... 40
D .............................................................................................. 40
Vě 26 ............................................................................ 40
Vě 27 B, .................................................................................. 41
*L 28 .................................................................................................................... 42
Vě 29 ............................................................................... 43
D ............................................................................................................ 43
D ů ............................................................................................................ 44
Vě 30 H ( , ) ........................................................................ 44
L 31 ě ..................................................... 44
*Vě 32 ů .................................................................................................... 45
Dů 33 ....................................................................................... 45
Vě 34 ............................................................................................... 45
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Z ......................................................................................................... 45
Vě 35 ........................................................................................................ 45
V ů: ................................................................................. 45
D ................................................................................................................................. 46
D ............................................................................................................... 46
D č .................................................................................................................. 46
Vě 37 ....................................................................................................... 47
Vě 38 ........................................................................................................ 47
Vě 39 ................................................................................................ 49
Vě 40 ............................................................................................... 50
Vě 41 .............................................................................. 51
H ě ............................................................................................................. 52
Vě 42 R ............................................................................................................................... 52
Vě 43 L, ř ě ......................................................................................... 52
Vě 44 ............................................................ 53
Vě 45 č ..................................................................................... 54
Vě 46 H ................................................................................................... 54
D ...................................................................................................... 54
K .............................................................................................................. 54
K ..................................................................................................................... 54
Př ..................................................................................................... 55
D ............................................................................................ 55
L 47 ............................................................................................. 55
Vě 48 ........................................................................................... 56
D č č ....................................................................................... 57
D ............................................................................................................... 57
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Vě 49 ........................................................................................... 58
Vě 50 č ................................................................................... 59
Půě ................................................................................................................................... 60
D ............................................................................................................ 60
T 51 ...................................................................................................................................... 60
Vř ůě ............................................................................................................. 62
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*
M ůů
*ř ů
*Nř ů
*Dů
*M
M
D :
č ů ů .
D , :
Ř, ⊂ ℝ , č ∈ ℝ ,
∈
≥ . T č
. A
. Ř, ⊂ ℝ , .ě 1 M
ě ž ,, … , . í: ∖ = ( ∖ )
∖ = ( ∖ )
M , :
P = 1:
∖
= ( ∖ )
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T:
∖ = ∖ P
n = n + 1:
CHCI:
( ∖ ) U:
∖ = ∖( ∩ ) = = ∖ ∪ ( ∖ ) =
P č ř, ů
= ( ∖ )
∪ ( ∖ ) =
S :
∪ =
T:
= ( ∖ )
ě 2 C
Nť , … , , , … , č. P :
≤
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ů ěě ě
P = ⋯ = = 0 í O ň
0.
Dů , č () č ( ), ě ř:
() = ( + ) P ř :
() =
+ 2
+
F . ř, ≤ 0. P:4
− 4
≤ 0
4
≤ 4
Č
M ř č
ℕ = 1,2,3,4,… M č
ℤ = ℕ ∪ 0 ∪ −; ∈ ℕ = …,−2,−1,0,1,2,… M č
ℚ = ; ∈ ℤ, ∈ ℕ Přč
= , ě ∙ = ∙
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č
M č ℝ ě , č (č + ∙), ř (č ≤), řč ň ř .
• V č .
• V ř č .
• A .
č :
• ∀, ∈ ℝ: + = + ( č )• ∀, , ∈ ℝ: + ( + ) = ( + ) + ( č )•
ℝ (č 0 ř ),
∈ ℝ
+ 0 = • ∀ ∈ ℝ ∃ ∈ ℝ: + = 0 ( . č č č , , č −)• ∀, ∈ ℝ: ∙ = ∙ ( )• ∀, , ∈ ℝ: ∙ ( ∙ ) = ( ∙ ) ∙ ( )• ℝ (. , č 1), ∈ ℝ 1 ∙ = • ∀ ∈ ℝ ∖ 0 ∃ ∈ ℝ: ∙ = 1 ( , č )• ∀, , ∈ ℝ: ( + ) ∙ = ∙ + ∙ ()
ř č :
• ∀, , ∈ ℝ: ( ≤ ∧ ≤ ) ⇒ ≤ ()• ∀, ∈ ℝ: ( ≤ ∧ ≤ ) ⇒ = ( )• ∀, ∈ ℝ: ≤ ∨ ≤ • ∀,, ∈ ℝ: ≤ ⇒ + ≤ + • ∀, ∈ ℝ: (0 ≤ ∧ 0 ≤ ) ⇒ 0 ≤ ∙
A :
B M . P č ∈ ℝ, :) ∀ ∈ : ≥ )
∀ ∈ ℝ, > ∃ ∈ : < ) T ∀ℰ ∈ ℝ, ℰ > 0 ∃ ∈ : < + ℰ ) č , ě č M
K
I M ě
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R č 1 3 č čě
P ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ K č
M č ů + , , ∈ ℝ.M č č. N ℂ č ň 1 ∙ = −1
ě ě
Nť ∈ ℕ, , … , ∈ ℂ, ≠ 0 . P + + + ⋯ + + = 0 ň ř ∈ ℂ.
D
B ⊂ ℝ. Č ∈ ℝ ň• ∀ ∈ : ≤ •
∀
∈ ℝ,
< ∃ ∈ : >
.
ě 3
Nť ⊂ ℝ . P M
Z
= −; ∈
M , A
M , A
T: ∃ inf = , :
∀ − ∈ : − ≥ ⇒ ≤ − Z :
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∀ℰ ∈ ℝ, ℰ > 0 ∃ − ∈ : − < + ℰ T:
∀ℰ ∈ ℝ, ℰ > 0 ∃ − ∈ : > − − ℰ
A M = −; ∈ , :∀ℰ ∈ ℝ, ℰ > 0 ∃ ∈ : > − − ℰ – ň ě , ,
sup = − = inf ∎
ě 4 č
P ∈ ℝ č č , . č ∈ ℤ , : ≤ < + 1
P = ∈ ℤ; ≤ C, ( ů ě 3 )S , N : = ∈ ℤ; ≤ T ℤ , ř ( ř 0 T
= inf (ℤ), ∈ ℝ
T ∀ ∈ ℤ: ≥ J
( − 1) ∈ ℤ − 1 ≥ ≥ + 1
, + 1 , SPOR!
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T ∃ sup = H ≤ − 1 ( + 1) > ⇒ ( + 1) ř ⇒ + 1 > ř, ě ≤ P
∀ ∈ ℝ ∃ ∈ ℤ: ≤ < + 1 ∎
ě 5 A
K ∈ ℝ ∈ ℕ ň < .
V = + 1, ℤ ℕ, = max +1,1 ∎
ě 6 ě
K ∈ 〈 0 ; +∞) ∈ ℕ ∈ 〈 0 ; +∞) ň = .
č, :
= , = , ≠ BNO < < ⇒ SPOR!P = 0 , č ≠ 0, ∈ ( 0 ; +∞) N , ∃! = P
= ∈ 〈 0 ; +∞),
≥
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(ň 1 ) T , :
= inf P > C , ( − ℎ) > , č .T:
0 < ℎ < min − ; N ň :
ℎ > 0 ℎ 0 U ( − ℎ):
( − ℎ)
= ( − )
−
+
=
= + ( − ℎ − ) ∙ (( − ℎ) + ⋯ + ) > A ( − ℎ) < ( − ℎ) > 0 :
> − ℎ ( + + ⋯ + ) = = − ℎ () >
P :
> − − ( ) = T:
( − ℎ) > P ( − ℎ) ∈ SPOR .P < C , ( + ℎ) < , č .
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T:
0 < ℎ < min 1; − ( + 1) N ň : ℎ > 0
ℎ < − ( + 1) ℎ < 1
U ( + ℎ):( + ℎ) = ( + ℎ) − + = = + ( + ℎ − )(( + ℎ) + ( + ℎ) + ⋯ + ) =
= + ℎ (( + ℎ) + ( + ℎ) + ⋯ + ) < A
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∈ ℚ ň < < C ě ∈ ℚ:
= , ∈ ℤ, ∈ ℕ
Z A ∃∈ ℕ , > , − > P ∈ ℤ , ∈ ( ; ), < = = < T :
č, čě ě
, č. T č . N
, č . ∈ ℝ ∖ ℚ ň < < V , ∈ ℚ, < < < V řč ě ě , √ , , 1. Č č .
= + 1√ 2 ( − ) N ∈ ( ; ) ⊂ ( ; ) ∈ ℝ ∖ ℚ ∎
L
D
J ř č řř č , ř, č. Č č .P , = ∈ ℕ. čů
∈ ℝ; ∃ ∈ ℕ: = D
Řě, :
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, čů
+
D
Řě, : , < ∈ ℕ +
D
+ = +
+ č
P č , = P ∈ ℝ , =
K
D
Ř, č ∈ ℝ , č ∈ ℝ, > 0 ř č ∈ ℕ , ř č ≥ | − | < , .:∀ ∈ ℝ, > 0 ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − | <
Řě,
, ,
∈ ℝ,
ě 8 č K č
→ = , = P:
P
lim = ⇔ lim( − ) = 0
B :
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lim→ = ∧ lim→ = Ú >
P . Z
<
,
∃ , ,
:
∀ > : | − | < ∀ > : | − | <
T ,
∀ > : ∈ ( − ; + )
∀ > : ∈ ( − ; + )
J = max ; , ě , ,
∀ > : − < < + ∧ − < < +
J < í : + < −, <
∎ ě 9 .
K .
V lim→ Z ∈ ℝ, > 0 Z , ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − | < T , č ě, − , +
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T < , čě (), = max| | + , ||, ||, … , ů , : ∀ ∈ ℕ: || ≤, ě .
∎
D
Nť č. Řě, ( ), ř č , = ∈ ℕ.
ě 10
Nť
. J
lim→ = ∈ ℝ, lim→ = .D :
∀ ∈ ℝ, > 0 ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − | < T čě > : = , A , , :
∀ ∈ ℝ, > 0 ∃ ∈ ℕ ∀ ∈ ℕ, ≥ > : − < ě 11
Nť lim→ = ∈ ℝ lim→ = ∈ ℝ , :() lim→( + ) = + () lim→() = () P ≠ 0 ≠ 0 ∀ ∈ ℕ, lim→ = () lim→( + ) = + C:
| + − − | < T :
| − + − | ≤ | − | + |b − | < 2 2
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() lim→() =
C:
| − | < V , ě − − :
| − + − | = | ( − ) + ( − ) | T :
≤ | ||( − )| + |||( − )| | − A | < | − B | < , :
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≤ 1 ( − A ) + AB ( − B ) ≤ 1
||| − A | + |A|
|B|| − B |
| − A | < | − B | < < 1|| + |A||B|
= | B| + |A||||B| Pě 2) ů , ∃ ∈ ℝ ∀ ∈ ℕ: b ≥ K
≤ | B| + |A|K|B| A
|||||| ∎
ě 12
Nť lim→ ť . P lim→ = 0 V, , :
∀ ∈ ℕ: || < N ě ( P VO2P
, ř )0 ≤ | | = | ||| ≤ || ∎
ě 13 ř
Nť lim→ = ∈ ℝ lim→ = ∈ ℝ ()
Nť ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : ≥ , ≥ () Nť < , ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : <
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() S
< 2) :
∃ ∈ ℕ ∀ ∈ ℕ, ≥ : < b A SPOR.
() Z
= ( ∀ ∈ ℝ, )∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − A | <
− A < < + ε ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − B | <
|B−| < B− <
> −
, :
= max , ∀ ≥ :
< + ε ∧ > − J :
< + − 2 ∧ > − − 2 < 2 + − 2 ∧ > 2 − ( − )2
< + 2 ∧ > + 2 T:
< + 2 <
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P:
<
∎
ě 14
Nť lim→ = lim → = ň :∃ ∈ ℕ ∀ ∈ ℕ, ≥ : ≤ c ≤ b
P:
lim→ = A V ∈ ℝ, > 0 V:
∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − A | < ⇒ > − ε ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − A | < ⇒ < + ε
P:
= max; ; N ∀ ≥ m :
− ε < ≤ c ≤ b < + ε T:
|c − A | <
∎ N
D
Řě, +∞ ( č), :
∀ ∈ ℝ ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : >
Řě, −∞ ( č), :
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∀ ∈ ℝ ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : < ě 8 č +∞ −∞. J lim→ , ř, , +∞. A −∞. J lim→ ∈ ℝ , ř, , lim→ = ∓∞
, ř, .
ě 9 . P ě 9.
ě 9
Nť lim→ = +∞ . P , .Nť lim→ = −∞ . P , .
ě 10 ( ) .
ě 13 ( ř) ě 14 ( ) .D ě 14.
ě 14
Bď ě .J
lim→
= +∞
∃
∈ ℕ ∀ ∈ ℕ, ≥
: b
≥ ,
lim→
= +∞
J lim→ = −∞ ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : b ≤ , lim→ = −∞ D ř
Rř ∗ = ∪/−∞;+∞ ř ř ℝ:
• < +∞ −∞ < ∈ ℝ, −∞ < +∞ • + (+∞) = ( +∞) + = +∞ ∈ ℝ∗ ∖ −∞ •
+ (−∞) = ( −∞) + = −∞
∈ ℝ∗ ∖ +∞
• ∙ (±∞) = ( ±∞) ∙ = ±∞ ∈ ℝ∗, > 0 • ∙ (±∞) = ( ±∞) ∙ = ±∞ ∈ ℝ∗, < 0 •
± = 0 ∈ ℝ.
Ně . J :(−∞) + (+∞), (+∞) + (−∞), (+∞) − (+∞), (−∞) − (−∞) (+∞) ∙ 0 , 0 ∙ (+∞), 0 ∙ (−∞), 0 ∙ (−∞) ,
,
,
,
∈ ℝ∗
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ě 11
Nť lim→ = ∈ ℝ lim→ = ∈ ℝ , :()
lim→( ± ) = ± ,
() lim→() = , () P ≠ 0 ≠ 0 ∀ ∈ ℕ, lim→ = ,
ě 15 ě
Nť lim→ = ∈ ℝ ∗, > 0, lim→ = 0 ∈ ℕ , ∈ ℕ, ≥ , > 0 . P:
lim→ = +∞
ě 16 ě
K .
• J , lim→ = sup ; ∈ ℕ • J , lim→ = inf ; ∈ ℕ
ě 17 B
Z ů .
ť :∃, ∈ ℝ ∀ ∈ ℕ: ≤ ≤
, , ň:1)
ℕ, ů ů
2) , , ∀ ∈ ℕ: ≤ 3) : ∀ ∈ ℕ: ≤ ≤ 4) (ě )
lim→ = lim→ lim→ − lim→ = 0 →
− = 0 (VOAL)
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, ě , ď .
V č , .V
〈; 〉 ě :
〈; + 2 ) ∪ 〈 + 2 ; 〉 P , ň ů čě čů . ( )T č 〈; 〉 Dů . Ič ř ě:
B 1) ∃ ∈ ℕ: ∈ 〈 ; 〉, > B 2)
= , < nebo < , = B 3)
≤ ≤
B 4) ř , ř, , → − = 0 − = − 2
P ěř:
B 1)
∃ ∈ ℕ: ∈ 〈 ; 〉
B 2)
= , < < , = B 3)
≤ ≤ B 4)
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− = − 2 č ( č č ů):
P, :∃ ∈ ℕ: ∈ 〈 ; 〉 ≤ ≤
− = − 2 = i.p.= − 22 = − 2
S:
V . I ě ů ř , ℕ K ů ř , ň
V ň ∀ ∈ ℕ: ≤ ≤ lim→ − = lim→ −
2 = 0
∎
D :
Nť , , ř, řř č:
•
: → • : → ()
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*č
*
Dč
D
G ; ∈ × ; ∈ , = () D
O
: → :
() = (); ∈ = ∈ ; ∃ ∈ :() =
*
*
*
*
* *B
*I
*I
F ě
D
F ě ( ) : → ℝ, č.
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D /
F : → ℝ , , ∈ , < , () < (). A ( , )
.
D
(. ) , , (. , ) .
D
Nť ⊂.Ř, •
, ∈ – ∈ () = −() • , ∈ − ∈ () = () • ∈ ℝ, > 0, ∈ ( + ) ∈ ,( − ) ∈ () = () = () • , č ∈ ℝ , ∈ () ≤ • , č ∈ ℝ , ∈ () ≥ • , č ∈ ℝ , ∈
() ≤
L
D
Nť ∈ ℝ > 0.P (, ) = ( − ; + ) (, ) = ( − ; +) ∖ . +∞ (. −∞) :(+∞,) = (+∞,) = ( 1 ⁄ ; +∞ ) (−∞,) = (−∞,) = ( −∞;−1 ⁄ )
D ě
Ř, č ∈ ℝ∗ ě, ∈ ℝ ∗, :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ∈ (, )
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ě 18 č
F ě ∈ ℝ∗. T č
lim→ () =
S:
lim→ () = , lim→ () = , ≠ N ∈ ℝ, > 0 , + < − ( ±∞ ± )P :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () (, ) ∧ ∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () (, )
Př + < − , : − < () < + < − < () < +
() < () SPOR!
∎ *D
*D
D
Ř, ě ∈ ℝ∗, lim→ () = () N:
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ((), ) ě 19
P ě ∈ ℝ∗, ∃ ∈ ℝ, > 0: (, )
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Z :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () (, ) T:
∃ > 0 ∀ ∈ (, ): B − < () < + ∎
*ě 20
()
lim→ () + () = +
() lim→ ()() = () lim→ ()() =
P ě
Nť ě ∈ ℝ. P + ě .P () ≠ 0 , ě .
ř ů ()
Z ř.: ( )
∈ ℝ, = +∞ C:
lim→ () + () = + ∞ = +∞ ∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () + () ∈ (+∞, )
N: ∀ ∈ ℝ ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () + () > ∈ ℝ Z :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ∈ (, ) ∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ∈ (+∞,)
Z č , ( ∀ ∈ ℝ, 1): ∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () > − 1
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A (( − + 1) ∈ ℝ , ∀ ∈ ℝ, ( − + 1)):∀ ∈ ℝ ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () > − + 1
N , ě : = min, P ů ř,
∀ ∈ ℝ ∀ ∈ (, ): () + () > − 1 + − + 1 = T:
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () + () ∈ (+∞,)
T.:
lim→ () + () = + Dů () :/
ů ():
V:
, ∈ ℝ, ∈ ℝ, > 0 C:
∃ ∈ ℝ, > 0 ∀ ∈ (, ): ()() ∈ ( , ) T:
()() − < B − − , ě , :
()() − = () − () () =
− + −
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34
( − ) + ( − )
≤ T ( ů ě):
≤ | | () − + | |() − ()|| = ≤ 1() () − + 1() | ||| () − =
T:
Př , , , ř () − ,() −
() ě č, , ě, ,
ř č . T č :
Z , :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ∈ (; ) T :
∃
∈ ℝ,
> 0 ∀ ∈ (,
): ()
∈ (;||2 )
Z č :
() > | |2 () < ||
D , :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ∈ (; )
N ( ):∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () − < Z č ( , , ě
∀ ∈ ℝ, > 0 || , ):∃ ∈ ℝ, > 0 ∀ ∈ (, ): () − < ∙ ||
Pě ( ):
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35
∃ ∈ ℝ, > 0 ∀ ∈ (, ): () − < ∙ | || | N ň:
= min ; ; A ď ř ( ě , 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): ()() − < ∎
ě 21 ě
Nť ∈ ℝ∗. Nť lim→ () = 0 , lim→ () = ∈ ℝ ∗ > 0. J > 0 , (,), lim→ ()() = +∞ .ě 22 , 2
Mě ∈ ℝ∗.()
Nť lim→ () > lim → ()ť. P (,) , :
∀ ∈ (, ): () > () () Nť (,) , ∀ ∈ (, ): () ≤ ().
Nť lim→ () lim→ (). P :lim→ () ≤ lim→ ()
()
( ) Nť ě (,) () ≤ ℎ () ≤
() ť
lim→
()
= lim→
(). P ě
lim→
ℎ()
ř .
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36
Nť ∈ ℝ∗, (,) lim→ () = 0 . Plim→ ()() = 0 Dů ()
Nť lim→ () = lim→ () = , ř > ∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ( , ) ∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () (, )
Z < | | , , . T. : + < −
K , : − < () < + < − < () < + T:
∃ ∈ ℝ, > 0 ∀ ∈ (, ): () < () ∎
Dů ()
P:
∀ ∈ (, ): () ≤ () lim→ () = lim→ () =
A :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ( , ) ∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () (, )
Z , ě ň: = min;
T :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ):
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37
− < () < + − < () < +
() ≤ ()
N :
< Z < | | , , . T. :
+ < −
J:
() < + < − < () ∧
() ≤ () A SPOR!
∎ Dů ()Z ∈ ℝ, > 0 P :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ( , )
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () (, )
Z , ě ň. = min;
Př :
() ≤ ℎ () ≤ () P:
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ):
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38
− < () ≤ ℎ () ≤ () < + ∎
ě 23
Nť ,, ∈ ℝ∗, lim→ () = , lim→ () = ě ň :
P) ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ≠ S) ě A
P : lim→( ∘ )() = P) , ě ( )
(S) Př, .Z ∈ ℝ, > 0 Vě ( A, ů ):
∃δ > 0 ∀ ∈ (A, δ): f () ∈ B(B,ε) Vř ( ě + , ε, δ):
∃δ > 0 ∀ ∈ P (c, δ): g() ∈ B(A,δ ) N , ,
g() , ř
f (). Dů , f () ů .Z δ , ě ň:
δ = min δ; δ P ů :
∃δ > 0 ∀x ∈ P(c, δ): ( f ∘ g )() ∈ B(B,ε) A , .
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39
(P) Př, ěZ ∈ ℝ, > 0 Vě ( ):
∃δ > 0 ∀ ∈ P (A, δ): f () ∈ B(B, ε) Vř ( ě, ů + , ε, δ):
∃δ > 0 ∀ ∈ P (c, δ): g() ∈ P(A, δ ) Z:
δ = min δ; δ P ů :
∃δ > 0 ∀ ∈ P(c, δ): ( f ∘ g )() ∈ B(B, ε) A č.
∎ ě 24 H ()
Nť ∈ ℝ∗, ∈ ℝ∗ lim→ f () = C . J ň ∈ ∖ ∈ ℕ lim→ = , lim→ () = .
V:
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0 ∀ ∈ (, ): () ∈ (,) ∀ > 0 ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : | − | <
N ( ):
∀ > 0 ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : (, ) A ∀ ∈ ℕ: ≠ :
∀ > 0 ∃ ∈ ℕ ∀ ∈ ℕ, ≥ : (, ) P ů :
∀ ∈ ℝ, > 0 ∃ ∈ ℝ, > 0: ∈ (, ) ⇒ () ∈ (, )
∎
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40
D
Nť , ∈ ℝ∗, < .P ř
(; )
∈ ℝ, < < .
ř 〈; 〉 ≤ ∈ ℝ, ≤ ≤ .ě 25
Nť , ∈ ℝ∗, BNO < . B (; ). P lim→ () lim→ (), řč :J (; ) , :
lim→ () = inf (;)
lim→ () = sup((;)) J (; ) , :
lim→ () = sup (;) lim→ () = inf((;)) F
D
Nť ⊂ ℝ ( čě ů). F : ⟼ ℝ , :• ř ě • ě , ř • ě , ř
ě 26
Nť , : ⟼ , : ⟼ ℝ, . P f ∘ g .
F , ě ... .
T ∈ , ∈ Z ě ě ( (S), ):
lim→ () = ∧ lim→ () =
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41
⇒ lim→( ∘ )() =
T ř , :
( ∘ )() = , , :
lim→ () = () = lim→() () = ( ∘ ) () =
T:
lim→( ∘ )() = ( ∘ ) () ∎
ě 27 B,
B 〈; 〉 ř, () < (). P ∈ ((); ()) ∈ ( ; ) , () = . N ( ůů L. P)
P :
= ∈ 〈 ; 〉, () ≤ V, ≠ ∅ (ě ) M ( ⊆ 〈 ; 〉)T . P
∃ sup = ∈ ℝ M, , ≤ ≤ ⇔ ∈ 〈 ; 〉
D , () = : () <
B ,
.
Pů ě 22 .
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42
lim→ () < lim→ ⇒
∃ > 0 ∀ ∈ (, ): () <
A , ě , ř , .
() > Z ě 22:
∃ > 0 ∀ ∈ (, ): () > T ě ,
, ř
, , , .V, ∈ 〈 ; 〉, :
= ⇒ () = () = ∈ (); () → ! = ⇒ () = () = ∈ (); () → !
P
∈ ( ; )
T , :
∀ ∈ (); () ∃ ∈ ( ; ): () = ∎
*L 28
Nť
⊂ ℝ :
∀, ∈ ∀ ∈ ℝ: < < ⇒ ∈ P .
(ů , ř , ě )
Nť ≠ ∅, = inf
= sup
∀ ∈ : ≤ ≤ (; ) ⊆ ⊆ 〈 ; 〉
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43
V ∈ ( ; ) T ∈ Z
∃ ∈ : > ∃ ∈ : < < <
⊆ 〈 ; 〉 V ∈ 〈 ; 〉 T ≤ ≤ T
∈ ( / )
ě 29
Nť J : → ℝ . P () .U, () ň L 28V
, ∈ ( ), < ⇒ ∃, ∈ : () = , () = V
∈ ℝ, () = < < = () Vě 27 (B, ):
∃ ∈ : () =
∈ () L 28 () .
D
Nť ⊂ ℝ, ∈ ň (. ⊂ ). Ř, ě , :
∀ ∈ : () ≤ ()
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44
B ě . S max čě , ě ( )+ min B ě č .
D ů
Nť ⊂ ℝ, ∈ ň (. ⊂ ). Ř, ě :
• , > 0 , : ∀ ∈ (, ) ∩ : () ≤ () • ,
> 0 , :
∀ ∈ (, ) ∩ : () ≥ () • , > 0 , : ∀ ∈ (, ) ∩ : () < () • , > 0 , : ∀ ∈ (, ) ∩ : () > ()
B , č .
ě 30 H ( , )
Nť ∈ . P ů ň lim = lim() = () L 31 ě
Nť ⊂ ℝ, ≠ ∅ = sup . P ů , lim→ = .
Z , č , ě
č. M
−
. T ,
, .∀ ∈ ℕ ∃ ∈ : > > − 1 T ě
lim→ = P = +∞ ě :
∀ ∈ ℕ ∃ ∈ : >
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45
*ě 32 ů
Nť 〈; 〉. P 〈; 〉 ě () ().
ů
Dů 33
B 〈; 〉. P 〈; 〉 .ě 34
B () J. P ()
()
ě 35
E (č ř ), :
() = ( 0;+∞) ()
log
(0;+∞)
() ∀, ∈ ( 0;+∞): log = log + log ()
lim→ = 1 ů:
1. log 1 = 0 log 1 = log 1 ∙ 1 = log 1 + log 1 = 2 log 1 2 log 1 = log 1 log 1 = 0 2.
∀ ∈ ( 0;+∞):log = − log
• ≠ 0, ů ř log 1 = log = log 1 ∙ 1 ∙ = log 1 + log 1 + log log 1 + log 1 + log = log 1 log 1 + log = 0 log 1 = − log 3.
∀ ∈ ℤ ∀ ∈ ( 0;+∞):log = ∙ log •
= 0 :
log = 0 ∙ log 0 = 0
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46
• ∈ ℕ :• 1. : log = 1 ∙ log log = log o č :
log = ( + 1) ∙ log log ∙ = log + ∙ log log + log = log + ∙ log log = ∙ log • ∈ −; ∈ ℕ
• = − • ∈ ℕ, ě, ∈ ℕ
4. lim→ log = +∞, lim→ log = −∞ • lim→ log = +∞
• L
•
lim→ log = −∞
lim→ log = −∞ lim→ − log = +∞ lim→ log 1 = lim→ log = +∞ 5. 6.
= ℝ 7. ě č ∈ ( 0;+∞) ň log = 1
D
D
Nť ∈ ℝ. P ě ě
′() = lim→ () − ()ℎ ě ě
() = lim→ () − ()
ℎ
ě ě
() = lim→ () − ()ℎ ř .
D č
Nť
ě
∈ ℝ. č
ě
,()
ř
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47
= ; ∈ ℝ ; = () + ′ ()( − ) ě 37
Nť
ě
∈ ℝ . P
ě .
CHCI :
lim→ () = () M , ě :
lim→ () − () = 0
lim→ () − ()
− ( − ) = 0
VOAL:
lim→ () − () − ∙ lim→( − ) = 0 A ( ℝ ), ů ě 22 ( ) , :
() ∙ 0 = 0 0 = 0
∎
ě 38
Př, ě ∈ ℝ ∈ ℝ.P :
() ( + )() = () + () () ()() = ∙ () () ( )() = ()() + ()() () () ≠ 0, :
() = ′ ()() − ()′()()
() () , .
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48
Dů () :
( + )() = lim→ ( + )() − ( + )()ℎ = lim→ () + () − () − ()ℎ
= = lim→ ()
− ()
ℎ ∙lim→ ()
− ()
ℎ = ′ () + ′ ()
Dů ():
S :
() = N ě, ()
Dů ():
Z :
( )() = lim→ ( )() − ( )()ℎ = lim→ () ∙ () − () ∙ ()ℎ = = lim→ () ∙ () − ()() + ()() − () ∙ ()ℎ = = lim→ () ∙ () − () + () ∙ () − ()ℎ = = = lim→ () ∙ lim→ () − ()ℎ
+ lim→ () ∙ lim→ () − ()ℎ
= () () , ě , :
= ′()() + ()′() Dů ():
, , ň () ≠ 0 , č 0 Z :
() = lim→
() − ()ℎ = lim→ ()() − ()()ℎ =
= lim→ 1() ∙ () ∙ ()() − ()()ℎ = = lim→ 1() ∙ () ∙ ()() − ()() + ()() − ()()ℎ = = lim→ 1
() ∙ () ∙ () ∙ () − () − () ∙ () − ()
ℎ =
= lim→ 1() ∙ () ∙ () − ()ℎ () − () () − ()ℎ =
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49
, .= = ′ ()() − ()′()()
∎ ě 39
Nť ě ∈ ℝ , ě ∈ ℝ = (). P:( ∘ )′() = ′ ()′()
Z
( ∘ )′() = lim→ (()) − (()) − = Cě :
= lim→ () − () − ∙ () − ()() − () =
= lim→ () − ()
() − () ∙() − ()
− =
= ′()′() J! Mů , () − () = 0 P
() = () − () − ≠ ′() =
F ě . T ě . A F ě , ů ř, : lim→ () = () ∧ lim→() () = ()
lim→ () = ′ () P:
( ∘ )() = lim→ () − () − = lim→ () ⋅ () − () − =
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50
= = lim→ () ⋅ lim→ () − () − = = ′() ⋅ ′ ()
∎ ě 40
Nť (; ) ě ∈ ( ; ) ′() ů . P ě = () :
()′() = 1
()
= 1
()
P :
() = , = () ()′() = lim→
() − () − = lim→ − () − () = 1 ′() Z () − () ů 0 T ě V39, :
() = () − () − ≠ ′() =
, ě V34 ( ), :
lim→
() = () lim→ () = ()
P V26 (VOSSF):
() = lim→ () = lim→ () − () () − =
A :
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51
= lim→ () − () () − () =
= lim→ −
() − () =
= 1lim→ () − () −
= ( )() T:
() = 1( )()
( )() = 1 () ∎
ě 41
Nť ě ∈ ℝ . J ′(), ′() = 0 .S ť
′()
′() ≠ 0
BNO ′() > 0 ( −′())P ě 22 ( ):
0 < ′() = lim→ () − () − ⇒
∃ > 0 ∀ ∈ (; ): () − () − > 0 N:
∈ (; ): − > 0 ⇒ () − () > 0 ⇒ () > () ∈ (; ): − < 0 ⇒ () − () < 0 ⇒ () < () N, (čě ) ř ě, ů ě , SPOR.
∎
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52
H ě
ě 42
Nť , ∈ ℝ, < :• 〈; 〉 • ( č ) ě ř (; ) • , () = ()
P ∈ ( ; ) ň ′() = 0 .N ř (1. ), (2. ) č ě (3. ) ň ř ě ( 0).
Bď ∀ ∈ 〈 ; 〉: () = (), ′() = 0 N ∃ ∈ ( ; ): () ≠ (). 〈; 〉 , ∈ 〈 ; 〉. P ě 41 ( ) ′() ď, . J ř 2 .
ě 43 L, ř ě
Nť
, ∈ ℝ, <
:
• 〈; 〉 • ( č ) ě (; )
P ∈ ( ; ) , : ′() = () − () −
N ř (1. ), č (2. ) ě ň ř ě ().
U ě ř ěě (R), ě , R.
D () ():() = () − () − () − ∙( −)
T:
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53
V ů () V ř :
() − () − ∙( −)
Oč , č () ň () ř .
Oěř , , () = ():() = () − () − () − ∙ ( − ) = () − () − () = ()
N ů R
:
∃ ∈ ( ; ): ′() = 0 T :
′() = ′ () − () − () − J ′() = 0 :
0 = ′() = () − () − () − () = () − () − ∎
ě 44
Nť
⊂ ℝ . Nť
ř ě
( ř ů č int ) .()
J ′() > 0 ∈ int , .() J ′() < 0 ∈ int , .()
J ′() ≥ 0 ∈ int , .()
J ′() ≤ 0 ∈ int , .
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54
ě 45 č
Nť ě ∈ ℝ lim→ ′(). P () :
() = lim→ () ě 46 H
Nť ∈ ℝ∗ lim→ ()(). Nť :• lim→ () = lim → () = 0 •
lim→() = +∞
P lim→ ()() :lim→ ()() = lim→
()() D
Nť
∈ ℝ . D
ě ě ′′() = lim→ ′() −
()ℎ .
Nť ∈ ℕ ∈ ℝ (č ()). P (+1) ě ě
()() = lim→ ()() − ()()ℎ .
K
K
K ů, ř č č.
M , ∈ ℝ ∈ 〈 0; 1〉. B ř :
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55
= + (1 − ) = + (1 − )( − ) ř
•
⟺ ě :• V řě ě ůě ů• V řě řč ě
= 12 + 12 = + 12 ( − ) • ⟺ , :
• V řě ě ůě ů
• V řě řč ě č = 34 + 14 = + 14 ( − )
D
Ř, • , ∀, ∈ ∀ ∈ 〈 0; 1〉 (()) ≤ () + (1 − ) () • ,
∀, ∈
∀ ∈ 〈 0; 1〉
(()) ≥ () + (1 − ) () • , ∀, ∈ ∀ ∈ 〈 0; 1〉 (()) < () + (1 − ) () • , ∀, ∈ ∀ ∈ 〈 0; 1〉 (()) > () + (1 − ) ()
L 47
F I ě , , ∈ , < < : () − () − ≤ () − () −
, ,
Z , , ∈ , < < N ∈ ( 0; 1) , ( ):
= + (1 − ) Oč , , ; ů ; () ; ()
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56
= ) + − ))
, ) ≤
P :
) − ) −
≤ − )
− =
A ů ) ) ř, ě č ě, :
= − )
− =
) − −
=
A
) ≤ :
= ) − −
≤ ) − )
−
T:
) − ) −
≤ ) − )
−
ď ě ě :
V č
C ( ): ) ≤
J , ) :
) − ) −
) − )
−
A SPOR.
ě 48
P ∈ ℝ ∗ < (; ) . P()
() > 0 ∀ ∈ ( ; ), (; ) ()
() < 0
∀ ∈ ( ; ),
(; )
()
() ≥ 0 ∀ ∈ ( ; ), (; )
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57
()
) ≤ ∀ ∈ ), )
Dů (),
E ), )
B 46, ř,
Nť
∈
P L (V43) :
∃ ∈ ): ) − )
− = ′)
∃ ∈ ): ) − )
− = ′)
Vě 44 ( 1. ): ∀ ∈ ): ) ≥ , . T:
⇒ ′) ≤ ′)
N ůč :
) − ) −
≤ ) − )
−
D č č
Nť ě ∈ ℝ č č ě; (). Ř, ; () č , :
() < () + ′ () ∙ ( − )
P č , ř, ; () č .D
Nť ′() ∈ ℝ . Ř, , Δ > 0 ,
• ∀ ∈ ( − ; ): ; () č ,•
∀ ∈ ( ; +
):
; () č
,
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58
• ∀ ∈ − ): ) č ,• ∀ ∈ + ): ) č .
ě 49
Nť ∈ ℝ . P () , .Uě ů ť () > 0 .
() = lim→ () − () −
/ č
∃Δ > 0 ∀ ∈ ( ; + ): () > ()∀ ∈ ( − ; ): () < () ň ř L ěV ∈ ( ; + ) L:
∃ ∈ ( ; ): () = () − () − () > ()
() − () − > () () − () > () ∙ ( − )
() > () + () ∙ ( − ) T ; () čV ∈ ( − ; ) L:
∃ ∈ ( ; ): () = () − () −
() < ()
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59
) − ) −
)
−) −) + ) ∙ − )
) ) + ) ∙ − )
T ) č
A SPOR .
Sě ě )
∎
ě 50 č
Nť ) ∈ ). Nť :
• ∀ ∈ ): ) ,
• ∀ ∈ ): ) .
P .
ř ∈ )
Z : ) ⇒ ) ⇒ )
)
V ∈ ), ň L, :
∃ ∈ ): ) = ) − )
−
)
)
) − ) −
)
) − ) ) ∙ − )
) ) + ) ∙ − )
č
∈ )
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60
Z : ) ⇒ ) ⇒ )
)
V ∈ ), ň L, :
∃ ∈ ):
) =
) − )
−
) )
) − ) −
)
) − ) ) ∙ − )
−) −) + ) ∙ − )
) ) + ) ∙ − )
č
∎
ůě
D
Př, ⟼ + , ∈ ℝ , +∞, : lim→ () − − = 0 −∞, :
lim→ () − − = 0
51
F +∞ ⟼ + , ě , :lim→ () = ∈ ℝ lim→ () − = ∈ ℝ
ř ů ⇒
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61
T:
→
)
= →
) − − + +
=
= →
) − −
+ +
=
V, ⟼ + :
→
) − − =
T VOAL:
= →
) − −
+ →
+
= + =
T:
→
)
=
A ě (VOAL):
→
) − = → ) − − + =
= →
) − − ) + → =
= + =
P:
→
) − =
ů ⇐
C :
→
) − − =
T VOAL:
→
) − − = → ) − − → =
J č , :
→
) − =
P
= →
) − − →
= − =
T:
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→
) − − =
A ⟼ +
∎
ř ůě
1)
Uč č .2)
Z : , , .3)
Dč č 4) Vř , č
. Uč .5) Sč č ,
. Uč .
6)
Uč .7)
Nč .