matan-1_2

x→x0

! ) &

C : , x		= 0 x		= 0: , 7		lim x	= lim x = 0 '
	n		n			n→∞ n	n→∞ n
lim f (xn) = lim sin πn = 0:					lim f (xn) = lim sin(2n + 1/2)π = 1
n→∞	n→∞				n→∞	n→∞
C 7 7					D		D <
, 5 M 7 7
7
LC "5"("'$" -4 ! Ox , x R <
	I		R: , x; δ<!

Oxδ	, x (x − δ, x +δδ)
	•	•
! Ox N δ<		Ox O , x
E , x
	$ :
% E 7 7 8
	(y0 = lim f (x)) :=
	x→x0
	• δ		ε
	( ε > 0 δ > 0 (x Ox0 ∩ dom f		f (x) Oy0 )).

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

: : :

, c R: , D x	dom f <
|f (x)| < c: f (x) < c: f (x)	> c ! , : :

M D E <

, x:

N O : N O

: N O

!L& #! ,

•

NTO ' y0 = lim f (x) Ox0 dom f

x→x0

#
NTTO ' ! y1	= lim f (x) y2 =	lim f (x)#
) y1 = y2	x→x0	x→x0

NTO C % E ( lim f (x) = y0) := ( ε > 0 δ >

0 x dom f (0 < |x − x0| < δ |f (x) − y0| < ε)).

: , 7

• δ	• δ	• •
y0 − ε < f (x) < y0 + ε x Ox0	∩ dom f Ox0	∩ Ox0 =Ox0 .

! ) * &
•
L |f (x)| < c x Ox0 : c = max{|y0 − ε|, |y0 + ε|}.
NTTO C	lim f (x) = y		)		( lim f (x) = y	2)
	(x x0	1			x x0
	→				→

: , D {f (xn)}:

, 7 ( lim f (xn) = y1) ( lim f (xn) = y2) !

n→∞ n→∞

y1 = y2

5 . 1 @ >9 & ' ;

LC "5"("'$" 3- " D y = f (x) y = g(x)

dom f = dom g = X: D

: <

: 7 7 7 8

		(f + g)(x) := f (x) + g(x), (f · g)(x) = f (x)g(x),
g			dom(f + g) = dom(f g) = X,
		(x) = g(x) , dom (f g) = X \ {x X : g(x) = 0}.
	f		f (x)

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

' ! # .

lim f (x) = y1

lim g(x) = y2

x→x0

x→x0

NTO

lim (f + g)(x) = y1 + y2;

x→x0

1 ·

NTTO

lim (f g)(x) = y

y

;

x

→

x0

2

f

y1 : y

NTTTO

lim

(

x

2

= 0

x x0

g

) = y2

→

:

C

{xn} X = dom f = dom g

, (x

= x

0

n

N)

( lim x

= x

) #

n

n→∞

n

0

( lim f (x

) = y

)

( lim g(x

) = y

) ! D

n→∞

n

1

n→∞

n

2

7 7

lim (f (xn) + g(xn)) = y1 + y2,

lim (f (xn)g(xn)) = y1y2,

n→∞

n→∞

	lim	f (xn)		y1	, y	2 = 0	.
		g(xn)	= y2
	n→∞

L {xn} <

7

! ) * &

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

' !

X

# . lim f

(

x

) =

y

1

< lim g

(

x

) =

x x0

x x0

→

• δ

→

y2 δ Ox0

x0

•δ

f (x) < g(x) x Ox0 ∩X

NTTO y = f (x) y = g(x) y = h(x) '

! X f (x) ≤ g(x) ≤ h(x)# .

lim f (x) = lim h(x) = y0 lim g(x) = y0
x→x0	x→x0		x→x0
NTO ! 7 7 y			R : , y		< y < y C
				1	•	δ1	2δ2
							•
% E 7 δ< Ox0							Ox0 : ,
						• δ1
	|f (x) − y1| < y − y1 x Ox0 ∩X,
	|g(x) − y2| < y2 − y					• δ2
				x Ox0 ∩X.

#

• δ

x Ox0 ∩X f (x) < (y − y1) + y1 = y = y2 − (y2 − y ) < g(x).

) δ = min{δ1, δ2}
NTTO " lim f (x) = lim h(x) = y :			ε > 0 <
→	→	0
x x0	x x0
• δ1	• δ2
δ< Ox0 Ox0		, x0 : ,
			• δ1
y0 − ε < f (x) < y0 + ε x Ox0 ∩X,
			• δ2
y0 − ε < g(x) < y0 + ε x Ox0 ∩X.
• δ1	• δ1	• δ
# D x Ox0	∩ Ox0	∩X :=Ox0 ∩X 7 7

y0 − ε < f (x) ≤ g(x) ≤ h(x) < y0 + ε,

|g(x) − y0| < ε: , % E E

("5 #!$" 3- lim f (x) = y1	lim g(x) = y2# .
x→x0	x→x0
•

Ox0 x0

NTO f (x) ≥ g(x) y1 ≥ y2=

! ) * &
NTTO f (x) > g(x) y1 ≥ y2=
NTTTO f (x) ≥ c c R y1 ≥ c=
NThO f (x) > c c R y1 ≥ c

: NTO 7 3 3 7 <

, 7 NTO NTTO + NTTTO NThO , NTONTTO g(x) = c

LC "5"("'$" 3 3 C y = g(x) z = f (x): , 7 im g dom f z = f ◦ g(x):

7 f ◦ g(x) = f (g(x)): 7 D

#"L " 3 2 y = g(x) z = f (x) 0

	im	g			dom f # ' ! lim g(x)			=	y	0
							x x0
lim f (y) = z0# .							→
y→y0
NTO y0 / dom f
NTTO f (y0) = z0:
lim f					◦	g(x) = z	0
	x		x0
	→

NTO y0 / dom f y0 / im g C {xn} dom f Q

: , xn → x0: xn = x0 n N # g(xn) → y0 g(xn) = y0 n N C M 7 f (g(xn)) → z0 !

{xn} , : ,

lim f ◦ g(x) = z0

x→x0

NTTO " z0 = f (y0): ε > 0 δ > 0 (y Oyδ0 ∩ dom f f (y) Ozε0 ) N) : ,

D , y0O

#	lim g(x) = y0: δ 7 δ > 0 :
• δ	x→x0
	δ	∩ dom f : im g dom f
, x Ox0	∩ dom g g(x) Oy0
	• δ	ε
L , 7: , x Ox0		∩ dom g: f (g(x)) Oz0

) " '$ L , NTO NTTO 7 3 2

! 7 7 : E , 5 M

7 7 f (y) = |WXYy|: g(x) = x sin x−1: x0 = y0 = 0 C , 7

lim x sin x−1			= 0: lim WXYy				|	= 1 L f	◦	g
x	→	0	y	→	0	|

, A 5 , 7

! ! - -. 7$ &

	xn =	1		x	=	2
		πn :				π(4n+1)
				n

5 5 B C ' > ;

L % E , , <

, , : D 7

C M 7 7

LC "5"("'$" 2-

(x

+

0

dom f (x > δ

NTO

lim f (x) = y

) := (

ε > 0

δ > 0

x

→ ∞

f (x) Oyε0 ));

0

{

n}

n→∞

n

∞

(x→+∞

) :=

(

dom f

=

NTPO

lim

f (x) = y

x

( lim x

+

lim f (xn) = y0));

n→∞

f (x) = y0) := ( ε > 0 δ > 0 x dom f (x < −δ

NTTO (x lim

→−∞

f (x) Oyε0 ));

(

) =

0) := ({

n} dom

(n

n

−∞

(x lim

y

f

=

NTTPO

f x

x

lim x

→−∞

→∞

lim f (xn) = y0));

n→∞

(x

0

ε > 0

| |

> δ

NTTTO

lim f (x) = y

) := (

δ > 0

x

dom f ( x

ε→∞

f (x) Oy0 ));

0

{

n}

n→∞

n

∞ n→∞

(

n) =

(x→∞

) := (

=

NTTTPO

lim f (x) = y

x

dom f ( lim x

lim f x

y0))

+C c'"'$" 5 M NTO NTPO: NTTO NTTPO: NTTTO NTTTPO 7 $ <

D M 7 7 , 7 7

7 5 7 7

D N 7 2O

LC "5"("'$" 2 3 δ<! ±∞ ∞ <

7 7	• δ	• δ
	O+∞= {x R : x > δ}, O−∞=

• δ

{x R : x < −δ}, O∞= {x R : |x| > δ}

# , % E $, , 7 " %8

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

! " % x → x0 Nx0 R x0 = ±∞, ∞O:

! ! - -. 7$ &

• δ

ε > 0 δ > 0 x , x dom f ∩ Ox0 (|f (x )

− f (x )| < ε).

#"L " 2- N % E O x0 R x0 =

±∞, ∞# lim f (x) '

x→x0

y = f (x) ! " % x → x0

C lim f (x) #

x→x0

• δ

ε > 0 δ > 0 x , x dom f ∩ Ox0 (|f (x ) − y0| < ε/2) (|f (x ) − y0| < ε/2).

L |f (x ) − f (x )| ≤ |f (x ) − y0| + |f (x ) − y0| < ε/2 + ε/2 = ε:

% E

C 7 : , % E

C

• δ

ε > 0 δ > 0 x , x dom f ∩ Ox0 (|f (x ) − f (x )| < ε).

7 7 {xn} dom f

: , xn = x0

n

N

lim x

n =

x

: 7: ,

{

f

(

x

n)}

n

→∞

0

D

!% E : ,

•δ

lim x

n =

x

:

δ >

N

N

n, m > N

x

, x

m Ox0 ))

n→∞

0

(

0

( n

: % E ε > 0 7 <

E N N : , D n, m > N |f (xn) − f (xm)| < ε

, : % E : ,

lim f (xn) = y0

n→∞

C 7: , : 7 <

7: D 7 7 7 7 M

{f (xn)}: {xn} dom f \ {x0}: lim xn = x0

n→∞

# % E 7 7: , n > N |f (xn) − f (xn)| < 1/N N ε = 1/N O L f (xn) − 1/N < f (xn) < f (xn) +

1/N C D N → ∞ , 7 7

% % E 7 : 7 7 : ,

7 7 <

7 7

! % 9 ' : &

LC "5"("'$" 2 4 " f : X → R 7 <

X R , ω(f, X) = sup |f (x ) − f (x )|

x ,x X

C$"

- ω(WXYx, [−1, 2]) = 2; 3 ω(|WXYx|, [−1, 2]) = 1;

• δ

2 ω(|WXYx|, O0) = 0

$ 7 <

% E 8

lim f (x))	(	ε >	0		δ >	0 (	ω	f,	• δ
( x x0							(		Ox0
→

) < ε)).

5 + " 9 & # ' E > ;

C , 7

lim sin x = 1,

x→0 x

7 7 7 , D : <

D 7 ! D $7 :

x (0, π/2) 0 < sin x < x <	sin x
		.	(4.1)
	cos x

C 7 M 7 8

(" 4- $ ! x	0		R ( lim sin x = sin x	lim cos x =
			x x0	0 x x0
cos x0)			→	→
5 7 ,			lim sin x = sin x0 C 7 <
7: , | sin x| ≤ |x| x			x→x0
		R 5 : N4-O

x (0, π/2) 7 7 0 < sin x < x: x (−π/2, 0) 7 7

, sin x 0 < sin(−x) < −x L | sin x| ≤ |x|:

|x| ≤ π/2 " |x| ≥ π/2: |x| ≥ | sin x|

: , | sin x| ≤ 1 < π/2 ≤ |x|

# : x0 = 0: lim sin x = 0:

<

{

x

n}

R

x→0

: lim x

n = 0

7 7

x

x

n| ≥ 0

\ {0}

n→∞

|

n| ≥ | sin

! 7 D <

, 7 lim

| sin

x

n|

lim

sin

x

n

= 0

n

→∞

= n

→∞

! % 9 ' : &

! 7 7 , x0

R #

{

x

n}

R

\ {

x

0}

: lim x

n

= x

: , 7

n→∞

0

|

sin x

n −

sin x

0

|

= 2

cos

xn + x0

sin

xn − x0

x

n −

x

.

2

2

≤ |

0|

! E 7

7

, 7 <

7

5 7: ,

lim cos x = cos x0 5 M 7 7: ,

x→x0

7 7 D

lim cos x = lim sin(π/2

−

x) =

x

x0

x

→

x0

→

= (π/2 − x = y, x → x0 y → π/2 − x0) =

= y

lim

x0 sin

y

= sin(

π/

2

−

x

0) = cos

x

.

→

π/2

−

0

C 7 7 , <

x < x < sin x , <

! N4-O 7 7

x

(0, π/2)

0 < sin

x

1

sin x

cos x

1 <

<

5 7 7 : cos x <

< 1 L 7 7:

sin x

cos x

x

, x (−π/2, 0) <, cos x , sin x C D

x → 0 77 7

D , 7 7

# 7

lim(1 + x)1/x = e.

x→0

' 7 7: , e = lim(1 + 1/n)n ! 7

D 7 7 D 7 7

{nk} {n} ( lim (1 + 1/nk)nk = e).

k→∞

7

{xk} <

: , (xk > 0) (klim xk = 0). 7 7

→∞

{nk = [1/xk]} L, : , M <

{n}

: , 7

nk+1 > 1/xk

≥

1

≥ xk

1

nk: 7 7

>

C M 7

nk

nk +1

1 + nk + 1

nk

≤ (1 + xk)1/xk < 1 + nk

nk +1

(4.2)

.

1

1

! % 9 ' : &

% 7 :

k→∞

1

nk

k→∞

nk

1

nk +1

1

−1

nk + 1

+ 1

nk + 1

lim

1 +

= lim

1 +

1 +

= e.

k→∞

nk

nk +1

nk

nk

nk

= k→∞

lim

1 +

1

lim

1 +

1

1 +

1

= e.

! N4 3O 7 7 D D <

, 7

lim (1 + xk)1/xk = e.

k→∞

# (xk

< 0) (klim xk = 0) C 7 yk = −xk #

→∞

lim yk = 0: yk > 0 5 :

k→∞

)1/xk = lim (1

)−1/yk

yk

1/yk

lim (1 + x

y

= lim

1 +

=

−

1 − yk

k→∞

k

k→∞

k

k→∞

yk

z

>

,

y

z

lim (1 + z )1/zk+1.

1 − yk

k →

= k

0

0 k

→ 0 = k→∞

k

C M 7

lim (1 + xk)1/xk = lim (1 + zk )1/zk (1 + zk) = e

k→∞

k→∞

<

E

L D 7 , D 7 D 7 $D

7 , D , E <

LC "5"("'$" 4- 5 y = f (x) y = g(x) <

	& x → x0 Nf g x → x0O:
x	•	f (x) = α(x)g(x): , 7 lim α(x) = 1
	Ox0	x x0
		→

C 7: , 7 M :

NTO f f x → x0;

NTTO f g x → x0: g f x → x0;

NTTTO (f g) (g h) x → x0: f h x → x0NTO L, : f (x) = α(x)f (x): α(x) = 1

! % 9 ' : &

NTTO C f g x → x0: f (x) = α(x)g(x):

•

lim α(x) = 1 , : , Ox0:

x→x0

α(x) = 0 ! M 7 7 g(x) = β(x)f (x): β(x) =

1/α(x): , 7 lim β(x) =	lim 1/α(x) = 1: g		f
→	→
x x0	x x0
x → x0

NTTTO C (f g) (g h) x → x0: f (x) = α(x)g(x): g(x) = β(x)h(x): , 7 lim α(x) = lim β(x) = 1 C M 7 f (x) =

x→x0		x→x0
γ(x)h(x): γ(x) = α(x)β(x): lim γ(x) = 1: f				h
x	→	x0
x → x0

#"L " 4- x sin x arcsin x VXx ln(1 + x) ex − 1 α−1((1 + x)α − 1) x → 0: α R \ {0}

# : , x sin x x → x0: <

7 , 5 7

				arcsin x	x arcsin x x → 0 :
	lim				= (arcsin x = y, x = sin y, (x	→	0)		(y	→	0)) =
				x
	x	→	0

lim SY(1 + x)

x→0 x

			= lim		y		= 1.
			y→0 sin y				x → 0 :
		x VXx
lim		VXx	lim		sin x		·	1	= 1.
		x			x
x	0		= x 0					cos x
	→		→					x → 0 :
	x SY(1 + x)

= lim ln(1 + x)1/x = ln	lim(1 + x)1/x	= ln e = 1.
x→0	x→0

N) 7 7 7 <

7 ,

lim ln x = ln lim = ln x0,

x0 > 0).

x→x0

x→x0

x ex − 1 x → 0 :

lim

ex − 1

= (ex = 1 + y,

x = ln(1 + y),

(x

→

0)

(y

→

0)) =

x

→

0

x