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Национальный минерально-сырьевой университет «Горный»

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8 4.3

1. ? lim f ( x) , f ( x) - ! , a

x a

& f ( x) ?

2.? ?

3.? & ?

4.L & !?

5.? «( ( »?

6.! & .

4.4. ( ! . ? ! ,

( H

" :

•( ! .

•# .

•? ! .

$ . # H &

, & [3], 2, . 53-56 –

M -( ( & $

( & ^ 2 ^ 61-70.

( !

a - ( ). L ,

f ( x) x a , ( " ! ,

, f (a) a ,

, &" f ( x) a .

. f ( x) (

a . @ f ( x) ! a ,

x a a , . .

lim f ( x) = f (a).

x a

! "

147

f (a 0) = f (a + 0) = f (a),

", ( (

f ( x) a .


5	!	,			f ( x)	x = a
$ , .
			sin x		, x 0,
%: 1. f ( x) = 5
%: 1. f ( x) = 5				x
		5		1, x = 0.
				1, x = 0.
, &				f ( x) x = 0 .
& '.	lim	sin x	= 1	(( ( ;		!
& '.	lim		= 1	(( ( ;		!
	x 0	x

$	!
& a = 0 +(x) = x ).	H x = 0 f ( x)

1. 7 x = 0

! ( , .

2. f ( x) =	2x	. , & f ( x)
	x 2
x = 2 .
	@ f ( x) x = 2 . >,
& '.

& ! ( .

I !.


1. #		f ( x)								a ,
"			a , (				f ( x)
, ( a .
2. #		f ( x) g ( x) a ,
) Cf ( x)		a ( C - );
&)	f ( x) + g ( x) a ;
)	f ( x) , g ( x) a ;
	f ( x)	a ,		g	(	a	)	0

) g ( x)									.

3. # u ( x)								a ,		f (u)

b = u (a) , f (u ( x))

a .

M , & !

( & .

7 .

148

. @ f ( x)

X ( &),

( ! .

. @ f ( x)

[a,b] , (a,b) f (a + 0) = f (a) , f (b 0) = f (b) .

5 , ( , $

& ; $ ,

# , + !

? ( 8 '). @,

, .

? ( 8 '). #

[a,b] , ( $

&$ .

E 4.5.

	O		[	x2	]	x
	O		a	x1	b
	.
				I. 4.5
? ( < '). #
	[a,b]
,		(			&	c ,
f (c) = 0 .

! 4.6.

B = f ( x1 )

C	[		x2 ] x
		x1 c
O a			b

A = f ( x2 )

I. 4.6

149

? ( < '). # f (x)

[a,b] x1, x2 [a,b]

f (x1 ) = A f (x2 ) = B , & & C , B

A B, ( & c (a,b) , f (c) = C . #. # f (x)

[a,b] , ! ( ( &

, B $ &$ .

? !

f (x) ( a ,

, & , a .

7 a ! f ( x) , ! (

f ( x) . E , a & ( ,

1)a ,

2)a , &

f (a 0) = f (a + 0) = f (a)

( & ,

f ( x)

a ).

( x) =

(

)

x(x2

+ 3) .

x = 0. >,

x = 0 - .

2. f ( x)

, x 1,

2x + 1, x > 1.

x = 1-

. 8 (

! ( .

f (1 0) = lim x2 = 1,

x 1 0

f(1 + 0) = lim (2x + 1) = 3.

x1+0

7 , x = 1- ! .

< ! .
. 7 a	f (x)	!
1-+ , & f (a 0)		f (a + 0)
" . I	f (a + 0) f (a 0) = 6a f
! f (x) a .

150

!,		f (a 0) = f (a + 0) ,	a	!
+	,	f (a 0) f (a + 0) ,			a
! ! + .					y
%. I					3
	x, x 0,				3
f ( x) =					x
x2 + 3, x < 0.					O
H		( ;0)	(0;+ ) .		I 4.7

f ( x) " ( ! (

( , , ( ! . I

x = 0 , ( ! ( .

		f (0 + 0)			= lim			x = 0,
					x	0+0
f	(	0 0	)	= lim		(	x2	+ 3	)	= 3.
	(		)	x	0 0	(			)

:& , , x = 0 - ! 1- . 7 , ! – , !

60 f = 0 3 = 3.

5 ! ( & . 4.7.

2. 7 a f (x) ! 2-+

, ( (

f (a 0) , f (a + 0) & ". H , ( (

f (a 0) , f (a + 0) &, a - !

! + .

%. f ( x) = 3x 2 . @ f ( x) x = 2 , , ! . 8 ( ! ( . M

( u( x) =				1	.
( u( x) =				x 2	.
	1			x 2
u (2 0) = lim	1		= ,
u (2 0) = lim	x	2	= ,
x 2 0	x	2
u (2 + 0) = lim	1		= + .
u (2 + 0) = lim	x	2	= + .
x 2+0	x	2
7
f (2 0) = lim 3u( x) = lim 3u = 0,
x 2 0	u
f (2 + 0) = lim 3u(x) = lim 3u = + .
x 2+ 0	u		+

O 2

I. 4.8

: &, ,

x = 2 - 2- , & . 5 ! ( & 4.8.

151

& ' !.

% 1.	2x 1, x 1,				:, &
	f ( x) =	3	,	x > 1.
		x

f ( x) x = 1.
& '. 8 ( f ( x)							x = 1.
				f (1 0)		= lim (2x 1) = 2 ,1 1 = 1,
				f (1 + 0)		x 1 0
				f (1 + 0)		= lim x3 = 13 = 1.
						x 1+0
H x = 1 , (
:	f	1 = 2 ,1 1 = 1			.
:		(	)		.
7			&,	&

! ( , , f ( x)

x = 1.

H " ( f ( x) ,

; ;

( .

		% 2.					f ( x)		=			2				.
		% 2.					f ( x)		=	1			2x			.						f ( x)
										1			2x
			y													& '.		@								,
											1 2x					= 0 , . .					x = 0 ,		, ,				x = 0 -
										.
								x		8 ( . 7 x																		0 0
			O					x			x < 0 , 2x											< 1 ,
			O								f (0 0)						= lim	2		= + .
																		2					1
																		2x					1
																	x 0 0 1	2x						2
	.		I 4.9														f	(0 + 0) = lim						2	= .
	.		I 4.9														f	(0 + 0) = lim					0 1 2x		= .
								x = 0 -													x	0+	0 1 2x
		>,						x = 0 -					2-													y
, & ( . 4.9).																									2
													cos x, x < 0,												2
							f ( x) = 52								, 0 x < 1, .
		% 3.					f ( x) = 52							x	, 0 x < 1, .												x
										5			2x, x > 1.														x
										4			2x, x > 1.												O	1	2
		& '. 8																							O	1	2
		& '. 8
(	;0		) ,	(	0;1			1;+					) ,					f	(	x	)
(			) ,	(	)			(					) ,						(		)
						" (											! (						.			I 4.10
(					,	,								!									.			I 4.10

152

f ( x) ,			&
x = 0 x = 1.
	8 ( ! .
1)	x = 0 .
	f (0 0)		= lim cos x = cos 0 = 1,
	f (0 + 0)		x	0 0
			= lim 2						= 2 , 0 = 0.
			= lim 2					x	= 2 , 0 = 0.
			x	0	+0
	:& , & (, , x = 0 -
1- , .
	M		60 f		= f (0 + 0) f (0 0) = 1
2)	x = 1.
	f (1 0)	= lim 2					= 2,
	f (1 0)	= lim 2				x	= 2,
	f (1 + 0)	x	1 0	(4 2x) = 4 2 ,1 = 2.
	f (1 + 0)	= lim		(4 2x) = 4 2 ,1 = 2.
		x	1+0
	H x = 1	f ( x)		, , !

, & & (, !

( . 4.10).

% 4.

f ( x) = 3

| x+1|

x+1

& '. 8 (

; 1

1;+

)

) (

" ( ! ( (. H

x = 1 . > x = 1 -

(

x + 1 , x < 1,

)

. 7 | x + 1|=

+ 1, x > 1,

(x+1)

lim f

( x) =

lim

x +1

1 0

3 .

f ( x) =

x+1

. I 4.11

lim

x+1

= 3 .

x 1+0

:& ,

, x = 1 -

( 4.11).

6 1 f = 3

= 2

8 | x 3 |

% 5.

f (x) =

x2 2x 3

& '. I . 7

f ( x) =

8 | x 3 |

( x 3)( x + 1)

N ,

x = 3 x = 1 -

f ( x) - . 8 ( ! .

x = 1 .

153

H x = 1 x 3 < 0 , , , | x 3 |= ( x 3) .

f ( 1 0) =	lim	8( x 3)	= lim	8	= + ,
f ( 1 0) =	lim	( x 3)( x + 1)	= lim	x + 1	= + ,
x	1 0		x 1
			(x< 1)
f ( 1 + 0) =	lim	8( x 3)	= lim	8	= .
f ( 1 + 0) =	lim	( x 3)( x + 1)	= lim	x + 1	= .
x 1+ 0		( x 3)( x + 1)	x 1	x + 1
			(x > 1)

M, x = 1 - & . 2) x = 3.

(

x 3

)

, x < 3,

| x 3 |=

x > 3,

8(x 3)

f (3 0) = lim

= lim

= 2,

( x 3)(x +1)

3 +1

x 3 0

x 3 0 x +1

(

x 3

f (3 + 0) = lim

)

= lim

= 2.

I 4.12

(x 3)( x +1)

x +1

3 +1

x 3+0

7 x = 3- .

63 f

= 2 ( 2) = 4 .

, ( .4.12.

	2x , x 0,
5			2	, 0 < x 2, .
% 6. f ( x) = ( x 1)				, 0 < x 2, .
5ln ( x 2), x > 2.
5

& '. N , $
x = 0 x = 2 . A !
1) x = 0
f (0 0)		= lim 2x = 20				= 1,
				x 0	0
f (0 + 0)		= lim			( x 1)2	= (0 1)2	= 1,
f (0) = 20				x 0+	0
f (0) = 20				= 1.
7 &
, x = 0 - .
2) x = 2
f (2 0)		= lim			( x 1)2	= (2 1)2	= 1,
				x 2	0
f (2 + 0)		= lim ln ( x 2) = .
				x 2+	0
7					& (, x = 2 -
& .

154

8 4.4

1.C ( .

2.( (, .

3.? ! (?

4.C ( (, (

5.( (,

6.C ( . .

7.C ( & .

4.5. % . B

( H "

•% .

•B .

•% , ! + .

•% , .

$ . # H &

, & [3], 3, . 58-65 –

y = f (x) 7 . H

-& x0 7 (

" 6x 0 , , x0 + 6x 7 . " x0 ,

" " 6x , &

6y = f ( x0 + 6x) f ( x0 ) .

I $ 66yx .

. # 6x 0 $ 66yx

& , !

155

y = f (x)				x		! x0		&
8	8
y , yx f '(x0 ) . A,
	y8 = f	8( x0 ) = lim	6y = lim		f ( x0 + 6x) f ( x0 )		.	(4.1)

		6x 0	6x	6x 0		6x
.				f '(x0 )				!

! , & (4.1). 7 &, ( & (

# y = f (x) " (

X , X ( ,

( x 7 (

! ( . 8 ! f &

f 8 .

% 1. 8 ( f (x) = x (

x .

f '(x) = x' = lim	(x + 6x) x	=	lim	6x	= 1.
	6x			0 6x
6x 0			6x

7 &, x ' = 1. % ,

(xn ) ' = nxn 1, n N.

(& [3]).

I + ! (.

(

(					y			l
Oxy l ,					y0 + 6y			l	M
Oxy l ,					y0 + 6y				M
"										T
y = f (x) ( .4.13).				7 &
( (									6y
! ( (			(					O	6y
M0 (x0 , y0 ) .
M0 (x0 , y0 ) .								dy
8 (	l	-						dy
8 (	l	-					M0
&					y0		M0		P
&					y0				P
M (x0 + 6x, y0 + 6y)								6x
" M0M ,			& "					6x
" M0M ,			& "
Ox (
.						0
.						0
C					O		x0	x0 + 6x		x
( (				l				I 4.13.
( (				l
M0 .
M0 .

156

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