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1. _______________________________ ȂĮșȘȝĮIJȚțȐ ȖȚĮ IJȘȞ īǯ ȁȣțİȓȠȣ _____________________________
ǼȊȀȁǼǿǻǾȈ Ǻǯ 118 (2020) IJ.2/51
ƶƼljǃ: ıŅ ƵǍLjĴNjĸƾķĶDžnj–ƣNjDžNJ–ƵǍLjƽǎĶDžĴ
ȀȣȡȚȐțȠȢ ȀĮȝʌȠȪțȠȢ, ǺĮıȓȜȘȢ ȀĮȡțȐȞȘȢ, ǹȞIJȫȞȚȠȢ ȂĮȖȠȣȜȐȢ
DZıțȘıȘ 1Ș
ǻȓȞȠȞIJĮȚ ȠȚ ıȣȞĮȡIJȒıİȚȢ g,h ȝİ IJȪʌȠȣȢ:
2
3x 30x 95
g(x)
4
(3x 5), R
4
O
O  țĮȚ
4x 5
h(x)
3
Į) ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș ıȣȞȐȡIJȘıȘ f g h
D
ȠȡȓȗİIJĮȚ ȖȚĮ țȐșİ š țĮȚ ȑȤİȚ IJȪʌȠ:
f(x) (goh)(x) 2
x 5x 10 x
O
ȕ) īȚĮ IJȚȢ įȚȐijȠȡİȢ IJȚȝȑȢ IJȠȣ ʌȡĮȖȝĮIJȚțȠȪ
ĮȡȚșȝȠȪ Ȝ ȞĮ ȣʌȠȜȠȖȓıİIJİ IJȠ ȩȡȚȠ:
x
lim f(x)
of
.
Ȗ) īȚĮ IJȘȞ IJȚȝȒ IJȠȣ ʌȡĮȖȝĮIJȚțȠȪ ĮȡȚșȝȠȪ Ȝ, ȖȚĮ
IJȘȞ ȠʌȠȓĮ IJȠ
of
x
lim f(x) İȓȞĮȚ ʌȡĮȖȝĮIJȚțȩȢ
ĮȡȚșȝȩȢ, ȞĮ ȣʌȠȜȠȖȓıİIJİ IJȘȞ IJȚȝȒ IJȦȞ ȠȡȓȦȞ:
o
4
x 2
f(x) 2x
lim
x x 14
țĮȚ
3
x
x
lim
f(x) x
of
KP
ȁȪıȘ
Į) ǼȓȞĮȚ g h
D D ƒ (ȖȚĮIJȓ;). ȉȠ ʌİįȓȠ ȠȡȚıȝȠȪ
1
A IJȘȢ g h
D İȓȞĮȚ:
^
1 h
x D :
$  `
g
h(x) D
 ^
4x 5
x : }
3
ƒ 
ȠʌȩIJİ ȠȡȓȗİIJĮȚ Ș ıȪȞșİıȘ țĮȚ ȑȤİȚ IJȪʌȠ:

3. g h (x) g h(x)
D
2
... x 5x 10 x f(x)
O
ȕ) ǼȓȞĮȚ
x
lim f(x)
of

4. 2
x
lim x 5x 10 x
of
O
x
... lim x
of 2
5 10
1
x x
§ ·
O
¨ ¸
¨ ¸
© ¹
(1). ǵȝȦȢ
x
lim x
of
f țĮȚ 2
x
5 10
lim 1
x x
of
§ ·
O
¨ ¸
¨ ¸
© ¹
1O
ȠʌȩIJİ įȚĮțȡȓȞȠȣȝİ IJȚȢ ʌĮȡĮțȐIJȦ ʌİȡȚʌIJȫıİȚȢ:
1. ǹȞ 1 0 1
O ! œ O IJȩIJİ ȜȩȖȦ IJȘȢ (1) İȓȞĮȚ
x
lim f(x)
of
f
2. ǹȞ 1 0 1
O œ O ! IJȩIJİ ȜȩȖȦ IJȘȢ (1) İȓȞĮȚ
x
lim f(x)
of
f
3. ǹȞ 1 0 1
O œ O IJȩIJİ
x
lim f(x)
of
=
x
lim
of

5. 2
x 5x 10 x
5
...
2
Ȗ) ȁȩȖȦ ȕ) ȖȚĮ 1
O İȓȞĮȚ

6. 2
f(x) x 5x 10 x
ȠʌȩIJİ: 4
x 2
f(x) 2x
lim
x x 14
o
=
0
2 0
4
x 2
x 5x 10 x
lim
x x 14
o
…

7. x 2 3 2 2
5(x 2)
lim
(x 2)(x 2x 4x 7) x 5x 10 x
o
5
124
ǼʌȓıȘȢ:
3
x
x
lim
f(x) x
of
KP
=
3
2
x
x
lim
x 5x 10
of
KP
=0
ȖȚĮIJȓ:
3
2
x
x 5x 10
KP
3
2
x
x 5x 10
KP
2
1
x 5x 10
d
ȐȡĮ
2
1
x 5x 10
3
2
x
x 5x 10
KP
d
2
1
x 5x 10
d
țĮȚ
2
x
1
lim
x 5x 10
of
§ ·
¨ ¸
© ¹
0
2
x
1
lim
x 5x 10
of
ȠʌȩIJİ Įʌȩ IJȠ țȡȚIJȒȡȚȠ ʌĮȡİȝȕȠȜȒȢ, ȑȤȠȣȝİ:
3
2
x
x
lim 0
x 5x 10
of
KP
.
DZıțȘıȘ 2Ș
ǻȓȞİIJĮȚ Ș ıȣȞȐȡIJȘıȘ
1 x
f(x) ln
1 x
.
Į) ȃĮ ȕȡİȓIJİ IJȠ ʌİįȓȠ ȠȡȚıȝȠȪ IJȘȢ ıȣȞȐȡIJȘıȘȢ
f.
ȕ) ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș ıȣȞȐȡIJȘıȘ f İȓȞĮȚ
ȖȞȘıȓȦȢ ĮȪȟȠȣıĮ.
Ȗ) ȃĮ ȕȡİȓIJİ IJȠ ıȪȞȠȜȠ IJȚȝȫȞ IJȘȢ f.
į) ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș ıȣȞȐȡIJȘıȘ f ĮȞIJȚıIJȡȑijİ-
IJĮȚ țĮȚ
x
1
x
e 1
f (x)
e 1
.
İ) ȃĮ ȣʌȠȜȠȖȓıİIJİ IJȠ
x 1
1
lim f(x)
f(x)
o
ª º
˜ KP
« »
¬ ¼
.
ȁȪıȘ
Į)
1 x
0
1 x
!
(1 x)(1 x) 0
œ ! 1 x 1
œ ȐȡĮ
ǹ= ( 1,1)
.

8. _______________________________ ȂĮșȘȝĮIJȚțȐ ȖȚĮ IJȘȞ īǯ ȁȣțİȓȠȣ _____________________________
ǼȊȀȁǼǿǻǾȈ Ǻǯ 118 (2020) IJ.2/52
ȕ) DzıIJȦ 1 2
x ,x $ ȝİ 1 2
x x
IJȩIJİ
1 2
1 x x 1
ȠʌȩIJİ 1 2
0 1 x 1 x
(1) țĮȚ
1
1
0
1 x
2
1
1 x
(2).
ȁȩȖȦ IJȦȞ (1),(2) İȓȞĮȚ: 1
1
1 x
1 x
ln x
2
2
1 x
1 x
Ÿ
/
1
1
1 x
ln
1 x
2
2
1 x
ln
1 x
Ÿ
1 2
f(x ) f(x )
Ÿ ȐȡĮ Ș f
İȓȞĮȚ ȖȞȘıȓȦȢ ĮȪȟȠȣıĮ.
ȈȤȩȜȚȠ:
Ǿ ȝȠȞȠIJȠȞȓĮ IJȘȢ f ȝʌȠȡİȓ ȞĮ ʌȡȠțȪȥİȚ ĮȞ
ȖȡȐȥȠȣȝİ IJȠȞ IJȪʌȠ IJȘȢ ıIJȘ ȝȠȡijȒ
1
f (x) ln 1
2 x
§ ·
¨ ¸
© ¹
Ȗ) ȁȩȖȦ IJȠȣ ȕ) țĮȚ įİįȠȝȑȞȠȣ ȩIJȚ Ș f İȓȞĮȚ
ıȣȞİȤȒȢ ȑȤȠȣȝİ:
f ( )
$
x 1 x 1
lim f (x),limf (x)
o o
§ ·
¨ ¸
© ¹
( , )
f f ƒ
(ȖȚĮIJȓ;)
į) ȁȩȖȦ IJȠȣ ȕ) Ș f İȓȞĮȚ ‘1–1’ ȐȡĮ ĮȞIJȚıIJȡȑijİIJĮȚ
țĮȚ ȜȩȖȦ IJȠȣ Ȗ) Ș 1
f : ( 1,1)
ƒ o . īȚĮ IJȠȞ IJȪʌȠ
IJȘȢ 1
f
șȑIJȠȣȝİ f (x) y țĮȚ ȑȤȠȣȝİ:
1 x
ln y
1 x
...
œ œ
x
x
e 1
x
e 1
ȐȡĮ
x
1
x
e 1
f (x) , x
e 1
ƒ
.
İ) ȁȩȖȦ IJȠȣ Ȗ) IJȠ
x 1 x 1
limf (x) lim f (x)
o o
f țĮȚ
f (x) 0
z ȖȚĮ xțȠȞIJȐ ıIJȠ 1 ȠʌȩIJİ ĮȞ șȑıȠȣȝİ
1
u
f (x)
IJȩIJİ
x 1
limu
o x 1
1
lim 0
f(x)
o
ȠʌȩIJİ:
x 1
1
lim f(x)
f(x)
o
ª º
˜KP
« »
¬ ¼ u 0
u
lim 1
u
o
KP
DZıțȘıȘ 3Ș
īȚĮ IJȚȢ įȚȐijȠȡİȢ IJȚȝȑȢ IJȠȣ Dƒ ȞĮ
ȣʌȠȜȠȖȓıİIJİ IJȠ
x
lim
of
g(x) ȩʌȠȣ:
2
2 x 3x 2
g(x) 4x x 1
2x 1
D
ȁȪıȘ
ȆȡȠijĮȞȫȢ Ș ıȣȞȐȡIJȘıȘ ȠȡȓȗİIJĮȚ ıİ įȚȐıIJȘȝĮ IJȘȢ
ȝȠȡijȒȢ ( , ), 0
N f N ! (ȖȚĮIJȓ;). ǼȓȞĮȚ
2
2
2
2
3 2
x
1 1 x x
g(x) x 4
1
x x
x 2
x
§ ·
D
¨ ¸
§ · © ¹
¨ ¸
§ ·
© ¹
¨ ¸
© ¹
= x
2
2
3 2
1 1 x x
4 x
1
x x 2
x
D
|x|=x
xof
2
2
3 2
1 1 x x
x 4
1
x x 2
x
§ ·
D
¨ ¸
¨ ¸
¨ ¸
© ¹
(1) .
ǵȝȦȢ
x
lim x
of
f țĮȚ
2
2
x
3 2
1 1 x x
lim 4 2
1
x x 2
2
x
of
§ ·
D
¨ ¸ D
¨ ¸
¨ ¸
© ¹
ȠʌȩIJİ
įȚĮțȡȓȞȠȣȝİ IJȚȢ ʌĮȡĮțȐIJȦ ʌİȡȚʌIJȫıİȚȢ:
1. AȞ 2 0 4
2
D
! œ D IJȩIJİ ȜȩȖȦ IJȘȢ (1) İȓȞĮȚ
x
lim g(x)
of
f.
2. AȞ 2 0 4
2
D
œ D ! IJȩIJİ ȜȩȖȦ IJȘȢ (1) İȓȞĮȚ
x
lim g(x)
of
f.
3. AȞ 2 0 4
2
D
œ D IJȩIJİ Ș g ȖȡȐijİIJĮȚ:
2
2 4x 3x 2
g(x) 4x x 1
2x 1
ȀȐȞȠȞIJĮȢ IJȘȞ İȣțȜİȓįİȚĮ įȚĮȓȡİıȘ
2
(4x 3x 2):(2x 1)
ȕȡȓıțȠȣȝİ ʌȘȜȓțȠ
1
2x
2
țĮȚ ȣʌȩȜȠȚʌȠ
3
2
ȠʌȩIJİ
2
4x 3x 2 1 3
2x
2x 1 2 4x 2
țĮȚ ȑIJıȚ
2
x x
1 3
limg(x) lim 4x x 1 2x
2 4x 2
of of
§ ·
¨ ¸
© ¹
1 1 1
0
4 2 4
ȖȚĮIJȓ:

9. 2 2
2
2
x x
4x x 1 4x
lim 4x x 1 2x lim
4x x 1 2x
of of
2
x
x 1 1
lim ...
4
4x x 1 2x
of
țĮȚ
x
3
lim 0
4x 1
of
. DZȡĮ:
x
1
, 4
4
limg(x) , 4
, 4
of
­
D
°
°
f D
®
°f D !
°
¯
.

10. _______________________________ ȂĮșȘȝĮIJȚțȐ ȖȚĮ IJȘȞ īǯ ȁȣțİȓȠȣ _____________________________
ǼȊȀȁǼǿǻǾȈ Ǻǯ 118 (2020) IJ.2/53
ȈȤȩȜȚȠ:
ȉȠ ȩȡȚȠ IJȘȢ g(x) ȣʌȠȜȠȖȓȗİIJĮȚ ȤȦȡȓȢ IJȘȞ ȕȠȒșİȚĮ
IJȘȢ įȚĮȓȡİıȘȢ ĮȞ ȖȡȐȥȠȣȝİ:
2
2 4x 3x 2
g(x) 4x x 1 2x 2x
2x 1
§ ·
¨ ¸
© ¹
DZıțȘıȘ 4Ș
DzıIJȦ ıȣȞȐȡIJȘıȘ f ıȣȞİȤȒȢ ıIJȠ 0, IJȑIJȠȚĮ ȫıIJİ
ȞĮ ȚıȤȪİȚ:f(x y)
f(x) y
VXQ f(y) x
VXQ ȖȚĮ
țȐșİ x,y
 . ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș f İȓȞĮȚ
ıȣȞİȤȒȢ ıIJȠƒ.
ȁȪıȘ
ĬİȦȡȠȪȝİ IJȣȤȩȞ 0
x  țĮȚ șĮ ĮʌȠįİȓȟȠȣȝİ ȩIJȚ Ș
f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ 0
x .
Ǿ ıȣȞȐȡIJȘıȘ f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ 0, ȐȡĮ
x 0
limf(x) f(0)
o
(1). ǼʌȓıȘȢ ĮȞ ıIJȘȞ ĮȡȤȚțȒ ıȤȑıȘ
șȑıȠȣȝİ x y 0 ʌȡȠțȪʌIJİȚ f (0) 0 (ȖȚĮIJȓ;),
ȐȡĮ
x 0
limf(x) 0
o
. īȚĮ țȐșİ h  ȑȤȠȣȝİ
0
f(x h)
= 0
f(x ) 0
h f(h) x
VXQ VXQ
İʌȠȝȑȞȦȢ 0
h 0
limf(x h)
o
= 0
f (x ) 0
h 0
lim h x
o
VXQ VXQ
0
h 0
limf(h) f(x )
o
(ȜȩȖȦ (1)) ȐȡĮ Ș f İȓȞĮȚ ıȣȞİȤȒȢ
ƒ.
DZıțȘıȘ 5Ș
ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș İȟȓıȦıȘ 3
x 3x 3 0
ȑȤİȚ
ȝȠȞĮįȚțȒ ʌȡĮȖȝĮIJȚțȒ ȡȓȗĮ, Ș ȠʌȠȓĮ ĮȞȒțİȚ ıIJȠ
įȚȐıIJȘȝĮ (2,3).
ȁȪıȘ
ĬİȦȡȠȪȝİ IJȘ ıȣȞȐȡIJȘıȘ
3
f(x) x 3x 3
,
x . Ǿ f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ țȜİȚıIJȩ įȚȐıIJȘȝĮ
[2,3] țĮȚ İʌȓıȘȢ f (2) f (3) 0
˜ (ȖȚĮIJȓ;), ȠʌȩIJİ
ıȪȝijȦȞĮ ȝİ IJȠ șİȫȡȘȝĮ Bolzano ȣʌȐȡȤİȚ U
IJȑIJȠȚȠ ȫıIJİ f ( ) 0
U . ȉȩIJİ Ș ıȣȞȐȡIJȘıȘ f
ʌĮȡĮȖȠȞIJȠʌȠȚİȓIJĮȚ ȦȢ İȟȒȢ:
f (x) (x )
U ˜ 2 2
(x x 3)
U U
Ǿ įȚĮțȡȓȞȠȣıĮ IJȠȣ IJȡȚȦȞȪȝȠȣ 2 2
x x 3
U U
İȓȞĮȚ 2 2
4 12
' U U 2
3 12
U
2
3(4 )
U
țĮȚ İʌİȚįȒ 2
U ! ȑʌİIJĮȚ 0
' . ǼʌȠȝȑȞȦȢ IJȠ ȡ
İȓȞĮȚ Ș ȝȠȞĮįȚțȒ ȡȓȗĮ IJȘȢ f.
DZıțȘıȘ 6Ș
DzıIJȦ Ș ıȣȞİȤȒȢ ıȣȞȐȡIJȘıȘ f ȝİ f(x)] ȖȚĮ
țȐșİ x . ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș ıȣȞȐȡIJȘıȘ f
İȓȞĮȚ ıIJĮșİȡȒ.
ȁȪıȘ
ǹȢ ȣʌȠșȑıȠȣȝİ ȩIJȚ Ș ıȣȞȐȡIJȘıȘ f įİȞ İȓȞĮȚ
ıIJĮșİȡȒ. ȉȩIJİ șĮ ȣʌȐȡȤȠȣȞ 1 2
x ,x  ȝİ 1 2
x x
țĮȚ 1 2
f(x ) f(x ).
z ȂİIJĮȟȪ IJȦȞ ĮȡȚșȝȫȞ 1
f (x ),
2
f(x ) ʌȠȣ İȓȞĮȚ ĮțȑȡĮȚȠȚ, șĮ ȣʌȐȡȤİȚ ĮȡȚșȝȩȢ Į Ƞ
ȠʌȠȓȠȢ įİȞ İȓȞĮȚ ĮțȑȡĮȚȠȢ. Ǿ ıȣȞȐȡIJȘıȘ f İȓȞĮȚ
ıȣȞİȤȒȢ, ȠʌȩIJİ ıȪȝijȦȞĮ ȝİ IJȠ șİȫȡȘȝĮ
İȞįȚĮȝȑıȦȞ IJȚȝȫȞ ıIJȠ įȚȐıIJȘȝĮ 1 2
[x ,x ] șĮ
ȣʌȐȡȤİȚ 0 1, 2
x (x x )
 IJȑIJȠȚȠ ȫıIJİ 0
f(x ) D, IJȠ
ȠʌȠȓȠ İȓȞĮȚ ȐIJȠʌȠ, įȚȩIJȚ D]. DZȡĮ Ș ıȣȞȐȡIJȘıȘ
f İȓȞĮȚ ıIJĮșİȡȒ.
DZıțȘıȘ 7Ș
ǹȞ ȖȚĮ IJȘ ıȣȞȐȡIJȘıȘ f ȚıȤȪİȚ
5 3
f (x) f (x) f(x) x
ȖȚĮ țȐșİ x ȞĮ
ĮʌȠįİȓȟİIJİ ȩIJȚ:
Į) f (x) x
d ȖȚĮ țȐșİ x .
ȕ) Ǿ f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ 0.
ȁȪıȘ
Į) Ǿ įȠıȝȑȞȘ ȚıȩIJȘIJĮ ʌȠȣ ȚıȤȪİȚ ȖȚĮ țȐșİ īȚĮ IJȘ
x ȖȡȐijİIJĮȚ: 4 2
f (x) (f (x) f (x) 1) x
˜ (1)
ȠʌȩIJİ f(x) 4 2
(f (x) f (x) 1) x
.
ǼʌİȚįȒ ȖȚĮ țȐșİ x ,
4 2
f (x) f (x) 1 1
t ,
ȑʌİIJĮȚ f(x) x
d ȖȚĮ țȐșİ x .
ȕ) ȁȩȖȦ IJȠȣ Į) ȚıȤȪİȚ: x f(x) x
d d ȖȚĮ țȐșİ
x .
ǼʌȓıȘȢ
x 0
lim x
o x 0
lim( x ) 0
o
,ȐȡĮ ıȪȝijȦȞĮ ȝİ IJȠ
țȡȚIJȒȡȚȠ ʌĮȡİȝȕȠȜȒȢ IJȠ
x 0
limf(x) 0
o
. ǼʌȚʌȜȑȠȞ ĮȞ
șȑıȠȣȝİ x 0 ıIJȘȞ (1) ʌĮȓȡȞȠȣȝİ f (0) 0 . DZȡĮ
Ș f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ 0.
DZıțȘıȘ 8Ș
DzıIJȦ ıȣȞİȤȒȢ ıȣȞȐȡIJȘıȘ f IJȑIJȠȚĮ ȫıIJİ
2
f(x) x
z ȖȚĮ țȐșİ x țĮȚ f(0) 1.
Į) ȃĮ ȣʌȠȜȠȖȓıİIJİ IJĮ ȩȡȚĮ
x
lim f(x)
of
țĮȚ
x
x x
lim
f(x)
of
KP VXQ
ȕ) ȃĮ İȟİIJȐıİIJİ ĮȞ Ș ıȣȞȐȡIJȘıȘ f İȓȞĮȚ 1–1.
Ȗ) ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș ıȣȞȐȡIJȘıȘ f ʌĮȓȡȞİȚ
İȜȐȤȚıIJȘ IJȚȝȒ.
į) ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș İȟȓıȦıȘ
2
f (x) x
x 1
+
f(x) x 2
0
x 2
ȑȤİȚ ȝȓĮ IJȠȣȜȐȤȚıIJȠȞ ȡȓȗĮ ıIJȠ įȚȐıIJȘȝĮ (1,2).
ȁȪıȘ
Į) ĬİȦȡȠȪȝİ IJȘ ıȣȞȐȡIJȘıȘ 2
g(x) f (x) x
,
x . Ǿ ıȣȞȐȡIJȘıȘ g İȓȞĮȚ ıȣȞİȤȒȢ țĮȚ ȖȚĮ țȐșİ

11. _______________________________ ȂĮșȘȝĮIJȚțȐ ȖȚĮ IJȘȞ īǯ ȁȣțİȓȠȣ _____________________________
ǼȊȀȁǼǿǻǾȈ Ǻǯ 118 (2020) IJ.2/54
x ȚıȤȪİȚ g(x) 0
z , ȐȡĮ įȚĮIJȘȡİȓ ıIJĮșİȡȩ
ʌȡȩıȘȝȠ țĮȚ İʌİȚįȒ g(0) 1 0
! , ȑȤȠȣȝİ g(x) 0
!
ȖȚĮ țȐșİ x ȠʌȩIJİ țĮȚ
2
f(x) x
! (1) ȖȚĮ țȐșİ
x . ǼʌȓıȘȢ 2
x
lim x
of
f , țĮIJȐ ıȣȞȑʌİȚĮ
(ȜȩȖȦ (1))
x
lim f(x)
of
f . īȚĮ țȐșİ x ȚıȤȪİȚ
2
f(x)
d
x x
f(x)
KP VXQ
d
2
f (x)
țĮȚ İʌİȚįȒ
x
2
lim 0
f (x)
of
, ıȪȝijȦȞĮ ȝİ IJȠ țȡȚIJȒȡȚȠ
ʌĮȡİȝȕȠȜȒȢ șĮ İȓȞĮȚ
x
x x
lim 0
f (x)
of
KP VXQ
.
ȕ) Ǿ ıȣȞȐȡIJȘıȘ f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ @
0,2 ȝİ
f( 2) 4
! ȜȩȖȦ (1),ȐȡĮ f (0) 1 2 f ( 2)
,
ȠʌȩIJİ ıȪȝijȦȞĮ ȝİ IJȠ șİȫȡȘȝĮ İȞįȚĮȝȑıȦȞ IJȚȝȫȞ
ȣʌȐȡȤİȚ

12. 1
x 2,0
 ȫıIJİ 1
f(x ) 2 . ǼʌȓıȘȢ Ș
ıȣȞȐȡIJȘıȘ f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ @
0,2 ȝİ f (2) 4
!
(ȜȩȖȦ (1)), ȐȡĮ f (0) 1 2 f (2)
, ȠʌȩIJİ
ıȪȝijȦȞĮ ȝİ IJȠ șİȫȡȘȝĮ İȞįȚĮȝȑıȦȞ IJȚȝȫȞ
ȣʌȐȡȤİȚ 2
x (0,2)
 ȫıIJİ 2
f (x ) 2 .
ǼʌȠȝȑȞȦȢ, ȣʌȐȡȤȠȣȞ 1 2
x ,x  ȝİ 1 2
x x
z țĮȚ
1 2
f(x ) f(x ) 2 ȠʌȩIJİ Ș ıȣȞȐȡIJȘıȘ f įİȞ İȓȞĮȚ 1–1.
Ȗ) īȚĮ țȐșİ x ȝİ x 2
! ȚıȤȪİȚ 2
x 4
! ȐȡĮ
ȜȩȖȦ IJȘȢ (1) țĮȚ f (x) 4
! . Ǿ f İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ
įȚȐıIJȘȝĮ [–2,2] ȐȡĮ ʌĮȓȡȞİȚ İȜȐȤȚıIJȘ IJȚȝȒ,
įȘȜĮįȒ ȣʌȐȡȤİȚ 0
x [–2,2] IJȑIJȠȚȠ ȫıIJİ
0
f(x ) f(x)
d ȖȚĮ țȐșİ x[–2,2].
ǼʌȓıȘȢ 0
f(x ) f(0)
d 1 4
, ȐȡĮ 0
f(x ) f(x)
d ȖȚĮ
țȐșİ x . ǻȘȜĮįȒ Ș f ıIJȠ 0
x ʌĮȓȡȞİȚ IJȘȞ
İȜȐȤȚıIJȘ IJȚȝȒ IJȘȢ.
į) Ǿ İȟȓıȦıȘ
2
f (x) x
x 1
f(x) x 2
0
x 2
ıIJȠ
įȚȐıIJȘȝĮ (1,2) ȖȡȐijİIJĮȚ ȚıȠįȪȞĮȝĮ
(x 2)
2
(f (x) x)
(x 1)(f (x) x 1) 0
ĬİȦȡȠȪȝİ IJȘ ıȣȞȐȡIJȘıȘ
h(x) (x 2)
2
(f (x) x)
(x 1)(f (x) x 1)
,
@
x 1,2
 . Ǿ ıȣȞȐȡIJȘıȘ h İȓȞĮȚ ıȣȞİȤȒȢ ıIJȠ [1,2]
ȦȢ ĮʌȠIJȑȜİıȝĮ ʌȡȐȟİȦȞ ıȣȞİȤȫȞ ıȣȞĮȡIJȒıİȦȞ.
ǼʌȓıȘȢ 2
h(1) (f (1) 1) 0
(ȖȚĮIJȓ;) țĮȚ
h(2) f(2) 3 0
! (ȖȚĮIJȓ;). DZȡĮ h(1)h(2) 0
țĮȚ
Įʌȩ Ĭ. Bolzano Ș İȟȓıȦıȘ h(x) 0 ȑȤİȚ ȝȓĮ
IJȠȣȜȐȤȚıIJȠȞ ȡȓȗĮ ıIJȠ (1, 2).
DZıțȘıȘ 9Ș
DzıIJȦ f : o
ȖȞȘıȓȦȢ ĮȪȟȠȣıĮ țĮȚ ıȣȞİȤȒȢ
ıȣȞȐȡIJȘıȘ. ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ Ș İȟȓıȦıȘ

13. f x x 0
ȑȤİȚ ȝȓĮ ĮțȡȚȕȫȢ ʌȡĮȖȝĮIJȚțȒ ȡȓȗĮ.
ȁȪıȘ
DzıIJȦ g(x) f (x) x
, x . Ǿ ıȣȞȐȡIJȘıȘ g
İȓȞĮȚ ȖȞȘıȓȦȢ ĮȪȟȠȣıĮ ıIJȠ (ȖȚĮIJȓ;) ȐȡĮ ȑȤİȚ IJȠ
ʌȠȜȪ ȝȓĮ ȡȓȗĮ. ǹʌȠȝȑȞİȚ ȞĮ ĮʌȠįİȓȟȠȣȝİ IJȘȞ
ȪʌĮȡȟȘ ȝȚĮȢ ȡȓȗĮȢ. ǹȞ ȚıȤȪİȚ f (0) 0 IJȩIJİ IJȠ
ȗȘIJȠȪȝİȞȠ ȚıȤȪİȚ. ȊʌȠșȑIJȠȣȝİ f (0) 0
! .
ȉȩIJİ g(0) 0
! .
ǼʌȚȜȑȖȠȣȝİ D ȝİ f (0)
D (1), IJȩIJİ 0
D ,
ȠʌȩIJİ f ( ) f(0)
D (2) (ȖȚĮIJȓ;). ǹʌȩ IJȚȢ (1) țĮȚ (2)
ȑʌİIJĮȚ f( ) 0
D D Ÿ g( ) 0
D . DZȡĮ g( )g(0) 0
D
țĮȚ Įʌȩ Ĭ. Bolzano șĮ ȣʌȐȡȤİȚ 1
x ( ,0)
 D ȫıIJİ
1
g(x ) 0 .
ȊʌȠșȑIJȠȣȝİ f (0) 0
IJȩIJİ țĮȚ g(0) 0
.
ǼʌȚȜȑȖȠȣȝİ ȕ ȝİ f (0)
E ! (3). ȉȩIJİ 0
E ! ȠʌȩIJİ
f ( ) f (0)
E ! (4). ǹʌȩ IJȚȢ (3) țĮȚ (4) ȑʌİIJĮȚ
f ( )
E E ! 0 g( ) 0
Ÿ E . DZȡĮ g(0)g( ) 0
E țĮȚ Įʌȩ
Ĭ. Bolzano șĮ ȣʌȐȡȤİȚ 2
x (0, )
 E ȫıIJİ 2
g(x ) 0.
DZıțȘıȘ 10Ș
DzıIJȦ f,g ıȣȞİȤİȓȢ ıȣȞĮȡIJȒıİȚȢ ȝİ ʌİįȓȠ
ȠȡȚıȝȠȪ IJȠ [0,1] ȝİ IJȚȢ İȟȒȢ ȚįȚȩIJȘIJİȢ:
x Ǿ f İȓȞĮȚ ȖȞȘıȓȦȢ ijșȓȞȠȣıĮ
x īȚĮ țȐșİ x [0,1]
 ȚıȤȪȠȣȞ 0 f(x) 1
d d țĮȚ
0 g(x) 1
d d
x f g g f
D D
ȃĮ ĮʌȠįİȓȟİIJİ ȩIJȚ ȣʌȐȡȤİȚ 0
x [0,1]
 IJȑIJȠȚȠ
ȫıIJİ 0 0 0
f(x ) g(x ) x .
ȁȪıȘ
DzıIJȦ h(x) f (x) x
, x [0,1]
 . Ǿ h İȓȞĮȚ
ıȣȞİȤȒȢ ıIJȠ [0,1], h(0) f (0) 0
t țĮȚ
h(1) f (1) 1 0
d . DZȡĮ ȣʌȐȡȤİȚ 0
x [0,1]
 IJȑIJȠȚȠ
ȫıIJİ 0
h(x ) 0 (ȖȚĮIJȓ;), ȠʌȩIJİ 0 0
f(x ) x (1). ĬĮ
ĮʌȠįİȓȟȠȣȝİ ȩIJȚ țĮȚ 0 0
g(x ) x .
DzıIJȦ 0 0
g(x ) x
! .
ȉȩIJİ, İʌİȚįȒ Ș f İȓȞĮȚ ȖȞȘıȓȦȢ ijșȓȞȠȣıĮ ȑȤȠȣȝİ:
0 0
f(g(x )) f(x )
(1)
0 0
g(f(x )) x
Ÿ
(1)
0 0
g(x ) x
Ÿ
ʌȠȣ İȓȞĮȚ ȐIJȠʌȠ.
DzıIJȦ 0 0
g(x ) x
.
ȉȩIJİ, İʌİȚįȒ Ș f İȓȞĮȚ ȖȞȘıȓȦȢ ijșȓȞȠȣıĮ ȑȤȠȣȝİ
0
f(g(x )) ! 0
f (x )
(1)
Ÿ 0 0
g(f (x )) x
!
(1)
Ÿ 0 0
g(x ) x
!
ʌȠȣ İȓȞĮȚ ȐIJȠʌȠ.
ǼʌȠȝȑȞȦȢ 0 0
g(x ) x .