matan-1_2

y

= lim = 1 y→0 ln(1 + y)

E

E

C M 7 , <

#"L " 4 3. f f1 g g1 x → x0

' & #

C (f f1) (g g1) x → x0: <

f (x) = α(x)f1(x) g(x) = β(x)g1(x)

, N4 2O # 7 7: , <

•

Ox0 : g1(x) = 0

β(x) = 0 ! M

L 7 7 , D <

D 7

5 / 9 ! o O

C M D , , < , : D 7 <

7 D E E 7 : 7

LC "5"("'$" 6- C x0 R x0 = ±∞: x0 = ∞ #

! 0 ; o O

+C c'"'$" 6- 5 M <

, % E

C 7 7 (

LC "5"("'$" 6 3 y = f (x)

x → x0: lim f (x) = 0 y = f (x) <

x→x0

! y = g(x) x → x0:

f (x) = α(x)g(x): α(x) Q , 7 x → x0 L <

, M 7 78 f o(g) N, F 7 FO

x → x0 " f o(g) g(x) Q , 7 x → x0:f 7

! g x → x0

C$" 6- x2 o(x) x → 0: x2 = x · x C$" 6 3 x o(x2) x → ∞: x = 1/x · x2

, E D 7 x

LC "5"("'$" 6 2 / 7 : , y = f (x)

! y = g(x) x → x0 N <

•

, f = O(g): , F E gFO: Ox0 c > 0

•

x Ox0 (|f (x)| ≤ c|g(x)|)

C$" 6 2 1/x = O(1/x2) x → 0: |1/x| ≤ 1/x2

|x| ≤ 1

C$" 6 4 1/x2 = O(1/x) x → ∞: 1/x2 ≤ |1/x|

|x| ≥ 1

+ 7 o O

#"L " 6- c R \ {0} x → x0

NTO o1(cf ) = co2(f ) = o(f ) O1(cf ) = cO2(f ) = O(f )=

NTTO o1(f ) + o2(f ) = o(f ), O1(f ) + O2(f ) = O(f ) o(f ) + O1(f ) = O(f )=

NTTTO o1(o2(f )) = o(f ) o1(O(f )) = o(f ), O(o1(f )) = o(f ) O1(O2(f )) =

O(f );

( ( 0 . .

&

NTO o1(cf ) = α(x)(c ·f (x)) = cα(x) ·f (x) = (c ·α(x)) ·f (x) = o(f ):

α(x): c · α(x) Q , 7 x → x0

! 7 6- < , +C c'"'$

#"L " 6 3 N M O f g

g(x)+α(x)g(x): lim α(x) = 0 : f (x) = (1+α(x))g(x) =

x→x0

β(x)g(x): lim β(x) = 1

x→x0

!7 7 4- 7 <

8

5 7 0 '

LC "5"("'$" 0- $ (x0, x0 + δ) N(x0 − δ, x0)O

N O δ! , x0

LC "5"("'$" 0 3 C x0 R

! 1 ,

y = f (x) x → x0

7 , 7 : 7 7 ∞ ±∞

+C c'"'$" 0- 5 M

% E D

+C c'"'$" 0 3 7 7 <

7 , 7 7

7

+C c'"'$" 0 2 7 % < E ,

,

C$" 0- : , y = WXYx ,<

7 L M

, 5 :

7 7 : , xn → 0 n → ∞:

7 : , 5 <

:

! 7 7 C {xn} dom f \

{x0} 7 {xn} = {xn} {xn}:

xn > x0: xn < x0 L, :

L 7

5 7 E 7

<

L E

: D D <

E

' 7 7: , y = f (x)

!' N !'O 7 7 X R:

x , x X: x < x 7 7 f (x ) ≤ f (x ) Nf (x ) ≥ f (x )O

7 <

D 7 D

#"L " 0 3 y = f (x)

! 3 ; ' ( &

! 7 0 3 7 7 M LC "5"("'$" 1- C x0 dom f y = f (x) <

LC "5"("'$" 1 3 "

, x0 dom f : 7 7 : , f

x0: , f x0

% E : 5 D y = D(x) <

, ! 7 7

M ,

C$" 1- 7 y = R(x):

D , D 5 : <

, x0 R: , Ox0 , N N: Ox0 , , ,

m/n: n < N +7 E Ox0 7 : 7

7: , : , 7 D D , : <

D N 7 : 7 : , x0: x0 QO

•

E N # 7 7: , x0 Ox0 |R(x)| < 1/N:

lim R(x) = 0

x→x0

(L% ( ' " !L& #! D 8

NTO , x0 R y = f (x)

, ;

NTTO y = f (x) y = g(x) , x0: D 77 (f + g)(x): (f · g)(x) , (f /g)(x) N

, g(x0) = 0O , x0 ;

NTTTO y = g(x) z = f (y) Q : , 7 im g dom f : g , x0: f , y0 = g(x0);

D 7 , x0

5 D <

,

LC "5"("'$" 1 2 : , <

7 : M 7

: , : <

: D 7 <

X R , 7 7 7 C(X)

7 7 E # '8 #"L " 1- #

' 7 7: , M 7 7 : , <

D: D: 7 , D: 7 <

, D D 7 , D 7 < , , 7 , D 7 ! ! <

7 y = ex y = ln x: ,

7 ' sin x cos x 7

E ' D 7 , D 7

7 ! M 7 , <

, D , 7 7 <

7 ! D D

7 1-

LD 7 7 <

7 D

LC "5"("'$" 1 4 # , x0 dom f ,

x0 dom f , : <

: f (x0 − 0) = f (x0 + 0) + 7

: <

Q

y = |WXYx| 7 , 7

+ 7 7 7 <

, C , 7 y = WXYx

5 D , 7

5 8 # ' ? > ;

#"L " =- N/ <% E 7 , 7 , O

(f C[a, b] f (a) · f (b) < 0) ( c (a, b) f (c) = 0).

5 7 [a, b] 7 " ,

: D D , D <

! 4 < ' ( &

7 , D

C : 7

7< E 7 , c (a, b): f (c) = 0: <

, 7 {In} D : D

7 C % E < % !c R (c Inn N) 7 7 {xn} {xn} <

("5 #!$" =- . y = g(x)

I a, b I g(a) = A = B = g(b)

! C ' A B c R

' a b g(c) = C

7 7 f (x) = g(x) − C ' [a, b] f (a) · f (b) = (A −C) ·(B −C) < 0 % 7 : f C[a, b]

) , : c (a, b) (0 = f (c) = g(c) − C)

#"L " = 3 N! E 7 7 7 , O C

# &

#

C f C[a, b] # x (a, b) Ox : , Ox ∩ [a, b]

f , D Ox <

[a, b] 7 :

/ < ( 7 , {Ox1 , . . . , Oxn }:

: 7 D mk ≤ f (x) ≤ Mk: k = 1, . . . , n ) , : x [a, b]

min{m1, . . . , mn} ≤ f (x) ≤ max{M1, . . . , Mn},

f , [a, b] C s = sup f (x)

x [a,b]

C 7: , f (x) < s x [a, b] # [a, b]

s − f (x) = 0 x [a, b]: D 7 7 , :

# 1/(s − f (x)) <

[a, b]: , C ,

$: xs [a, b] (f (xs) = s)

, : i = infx [a,b] f (x) 7