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Национальный минерально-сырьевой университет «Горный»

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P :

1.P ' " 9 / 0 - "

/ 0.

2.! f (x) - / 0, '

9, / 0 g(x)

- " , , / 0 f ( x) , g(x)

-" .

3.! x a / 0 f ( x) -

, / 0 g ( x)

" "- x

a ,

/ 0 f ( x) , g(x) - " "- x

a .

! / 0 f ( x) - " a

, , / 0

f (x)

" "- a ; f ( x) - " "-

- a ,

- " .

f (x)

11. > a) lim

cos x

;

lim

sin x

+ ,.

x 1 x 1

x2 + 3

) x

1 / 0 ( x 1)

- " , ,

x 1

" "- , cos x

cos1 0, , cos x

cos x

" "- , . . lim

= .

x 1

/ 0

x2 + 3

- " "- , ,

+ 3

sin x

" . E 0

- ', ,

sin x

" , . . lim

sin x

= 0.

x2 +

x2 + 3

! / ' '

00 , , 0 , , , 00 , 0 ,1 ,

196

+ / 0 " "-

* 12. > lim	x3 1	.
	3x + 2
x 1 x2

+ ,. > x = 1 -

00 . M + +

", " + ' ". H + /:

(a3 b3 )= (a b)(a2 +ba + b2 ), . . x3 1 = (x 1)(x2 + x + 1).

" . ! -

x1, x2 , +

": ax2 + bx + c = a(x x1 )(x x2 ).

x1 = 1, x2 = 2,

x2 3x + 2 = (x 1)(x 2). = ",

lim

x3 1

= lim

(x 1)(x

2 + x +1)

= lim

x2 + x +1

= 3.

3x + 2

(x 1)(x 2)

x 2

1 x2

x 1

13. > lim

1 2x2

x 0

+ ,. > x = 0,

. T + " +, + ,

(

+ 1):

1 2x2

(

1)(

+1)

1 2x2

lim

1 2x2

= lim

2x2

1 2x2

= lim

x 0

x2 (

1 2x2 + 1)

x 0 x2

( 1 2x2 + 1)

= lim

2x2

= lim

= 1.

+ 1)

x 0 x2

( 1 2x2

x 0 1 2x2

+ 1

C " / (a b)(a + b) = a2 b2.

C. ! 0 -

", " + +, +

, +, + .

* 14. > lim	3x3	2x + 5	.
		+ 3x 8
x 7x2

197

+ ,. ,

. "

x3 , x2

( - x + ' '-

5 "

2x + 5

lim

= lim

= lim x$

2 !

8 "

x 7x2 + 3x 8

7 +

$ 7 +

x2 '

1x , x12 , x13 , " " "- , - " , ,

, + " 73 . x - " "- ,

, x , 73 + " "- ,

lim 3x3 2x + 5 = . x 7x2 + 3x 8

: ' +

0 / 0.

* 15. >	lim	5x + 8	.
* 15. >	lim		.
x	+	4x2 3
+ ,.

5 +

5x +

x$5 +

lim

+ 4x2 3

3 "

$ 4

= x

+ , x > 0 , ,

= x . ,

8 "

x$5 +

5 + x

5 + 0

lim

x '

lim

x '

lim

x +

+ x 4

4 0

lim (

+ x).

* 16. >

x2 + 2x

+ ,. F ( ). T +

+ +:

198

(

+ x)=

(

+ x)(

x)=

x2 + 2x

lim

x2 + 2x

lim

x2 + 2x x

x2 + 2x x2

lim

x2 + 2x x

. M ":

lim

= lim

lim

x x2 + 2x x

1 +

= x

, x < 0 , ,

= x . = '

lim

x x 1+

lim

% = 1.

+1%

; & "

1 &

E 0 +(x)

3( x) , " x

a , ,-

lim

+(x)

= 1. J " "

3 (x)

: + ( x)

~ 3( x)

a . + -

: - " , , " ,. !

x a + ( x) - " , lim +( x) = 0,
	x a
sin + ( x) ~ + ( x);	ln (1+ +( x)) ~ + ( x); e+( x) 1 ~ + ( x);
tg+ ( x) ~ + ( x); a+(x) 1 ~ + ( x) ln a; n			1 ~	+ ( x)
		1+ + ( x)			;
				n

199

- /-

-+ ( x).2

arcsin + ( x)

~ + ( x); 1 cos +x ~

; arctg+ (x) ~ + (x).

* 17. > lim

(e3x2 1)sin 2x

3x)(1 cos 2x)

x 0 ln(1

( 3x) 0,

+ ,. = x

3x2

. C " ,

4x2

e3x2

1 ~ 3x2 ;

sin(2x) ~ 2x ;

ln(1 3x) ~ ( 3x);

1 cos 2x ~

(e3x2 1)sin 2x

3x2

= '

lim

= lim

, 2x

= 1.

x 0 ln(1 3x)(1 cos 2x)

0 ( 3x), 2x2

-" &

. @ &

! / 0 y = f (x)

a , a / 0

1)x = a / 0 f (x) ;

2)x = a / 0 f (x) ,

" :

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

' f (a 0) f (a + 0)

0 f a .

! , f (a 0) f (a + 0) , x = a -

' . , f (a 0) f (a + 0),

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

! "

", a ' ( -

" ' , "

- ").

, / 0 + " -

200

* 18. > / 0 y = f (x), -

. H ' / 0.

' /.
	x3 , x 0,
f (x) =	53x , 0 < x 1,

	5 2x + 5, x > 1.

+ ,. + ' + ( ;0), (0; 1) (1; + ) /- 0 f (x) , / 0. P,

+ ' , + / 0 f (x) " , -

' " 0 , +, x = 0

x = 1.

> , :

H x = 0 :

x3 =

f (0 o) =

lim

f (x) =

lim

0 o

f (0 + o) =

lim

f (x) = lim

= 30 = 1.

0+o

@" , + -

", , x = 0 I . -

O 1

x = 0

/ 0

f (x)

2,5

0 = f (0 + o) f (0 o) = 1 0 = 1.

M x = 1.

M. 4.1.9

lim

f ( x) = lim 3x = 31 = 3;

x 1 o

lim

f (x) = lim ( 2x + 5) = 2 ,1+ 5 = 3;

x 1+o

1+o

(1) = 31 = 3.

@ / 0 -

, , , / 0

f (x) . < / /-

0 " + . 4.1.9.

x + 1

* 19. > / 0

f (x) =

x2 2x

, ' / 0, -

' / .

+ ,. " ":

x + 1

f (x) =

x2 2x

(x +

1)(x 3)

E 0 x = 1 x = 3 , , ,

. > :

201

1. H

x = 1

1 o

x + 1 < 0

x + 1

= (x + 1). P,

f ( 1 o) =

lim

(x + 1)8

lim

1 o (x + 1)(x 3)

1 o x

: '

x +1

f ( 1

+ 0) =

lim

1)( x 3)

1+o (x +

, ,

= 2.

8(x +1)

= lim

lim

= 2.

+1)(x 3)

x 1+o ( x

x 1+o

= " , x = 1 -

' .

, , - I .

6 1 = f ( 1+ o) f ( 1 o) = 2 2 = 4 -

/ 0.

x = 3

M. 4.1.10

x +1 > 0 , , | x +1|= x +1 ,

8( x +1)

f (3 o) = lim

x +1

= lim

= .

( x +1)( x 3)

+1)(x 3)

x 3

x 3 o

x 3 o ( x

x 3 o

f (3 + o)

= lim

x +1

= lim

= + .

+1)(x 3)

x 3

x 3+o (x

x 3+o

= ", x = 3 - " ' ' . < /

/ 0 . 10.
		* 20.			>		/ 0
		1
	f (x) = 3			, , ,-
	f (x) = 3		2 x	, , ,-				y
' / / 0 .
		+ ,. H , / 0
					x = 2 , ,			O 1
, .					> ,			O 1
, at					+ t	+ at	0	2	x
t , a > 1. x						2 o 2 x > 0		M. 4.1.11
1			+ ,			,
	2 x		+ ,			,
	2 x

f (2 0) = lim 32 x = + .

x 2 0

= ", x = 2 / 0 " '

202

			x			2 + o						2 x < 0												1								,							,
			x			2 + o						2 x < 0											2 x									,							,
							1																2 x															1
							1
f (	2 + 0) = lim 3										= 0.																					x				±								0
									2 x								C,
									2 x								C,																							2 x
	1			x		2+0																																		2 x
	1		= 1,												y = 1 ' .
lim 3		2 x	= 1,		. .										y = 1 ' .
x
J ' / / 0 " + . 4.1.11.
													*
	@ // 0:
	! / 0 u(x) v(x) // 0 x , ,
:													8							8
						1.							8			8		8	2.	8										8	(c = const);
						1.	(u + v) = u										+ v ;		2.	(cu)				= cu							(c = const);
							8								8			8		8								8			uv			8
							8								8			8		! u "							u v				uv
						3.		(uv)					= u v + uv ; 4.							$		%		=											.
						3.		(uv)					= u v + uv ; 4.							$		%		=						v		2			.
																				& v '										v
															= " 0 :
1.	(c)8 = 0;														2.			(xm )8 = mxm 1																	3.			(sin x)8					= cos x;
			8															8		1																		8						1
4.	(cos x)			= sin x;											5.			(tgx)	=							;									6.			(ctgx)					=			;
4.	(cos x)			= sin x;											5.			(tgx)	=		cos2 x					;									6.			(ctgx)					=	sin2 x		;
				8			1													8										1													8	1
7.	(arcsin x)					=								;	8.			(arccos x) =															;		9.			(arctgx) =								;
																																													1+ x2
										1 x2																			1 x2																1+ x2
10.	(arcctgx)8 =									1			;		11.			(ex )8 = ex ;																	12.				(ax )8 = ax ln a;
10.	(arcctgx)8 =						1+ x2						;		11.			(ex )8 = ex ;																	12.				(ax )8 = ax ln a;
							1+ x2																																		(a>0, a 1)
																																									(a>0, a 1)
13.	(ln x)8			=	1 ;										14.			(loga x)8 =							1					.
13.	(ln x)8			=	1 ;										14.			(loga x)8 =					x ln a							.
						x																	x ln a
																			(a>0, a 1)
																																				5					x2
																																				5					x2
		*		21. > / 0 y = 23 x																																			+			+ 3.
		*		21. > / 0 y = 23 x																																	x2		+		2	+ 3.

+ ,. F // 0

203

							8	!		2	"	8			1			8		8				8

	3	8	!	5 "					x					!		3	"		2		1		2
	3	8	!	5 "				$	x		%			!		3	"		2		1		2
y8	= (2	x )	$		2	%		+ $	2		%		+ (3)8	= 2$ x			%	5(x		) +	2	(x		) + (3)8.
			& x		'			&	2		'			&			'				2

H / + / 0 (" / N 2):

	8	1	(x)	1	1		( 2) x	2 1 1 2 1	2 2	3		3	2
				3

y		= 2 3				5		+ 2 2x + 0 = 3 x			+10x		+ x = 33 x2

* 22. > / 0 y = (cos x + 5)e x .

+ ,. F // 0

/ N 4 N 11:

y8 = (cos x + 5)8 e x + (cos x + 5)(e x )8 =

= ( sin x + 0)e x + (cos x + 5)e x = e x (cos x sin x + 5).

* 23. > / 0 y = arctgln xx .

+10x3 + x.

+ ,. F // 0 ' " / N 9 N 13:

ln x arctgx ,

1+ x

y8 =

(arctgx) ln x arctgx(ln x)

ln2 x

(

)

x ln x

1+ x2

arctgx

(

+ x2

)

ln2 x

x 1

+ / 0 y = f (u(x))

y8x = fu8 (u)u8x .

= , " + / 0, + //-

0 " - " / 0 + ' u ,

" ' u " , ' +

/ 0 u x .

J + / 0, " '

' // 0 / 0.

* 24. > / 0 y = ln(1 + 2 cos x).

+ ,. H / 0 - +, + ' u = (1+ 2 cos x). P '

204

y8 = (ln u)8u u8x =

+ 2 cos x)8

( 2sin x).

1 +

2 cos x

* 25. > / 0 y =

8 + sin 2 x

+ ,. H +

/ 0

y = f (u(v(x))), ' f (u) =

u(v) = 8 + v2 ,

v = sin x. -

// 0 + / 0 ( // 0 " --

" / 0 f ):

8	8		8	8	8			1					2			8		1			8
8	8	=	8	8	8	=				(8	+ sin		2	x)			=				8		=
fuux		=	fuuvvx			=			u	(8	+ sin			x)			=				2vvx		=
						2			u						x		2		8 + sin2 x
=			2sin x				(sin x)8				=	2sin xcos x							=	sin 2x		.
=							(sin x)8				=								=			.
	2		8 + sin2 x								2		8 + sin2 x					2		8 + sin2 x
				'		&

* 26. > Oy Ox -

y = f (x), ' f (x) =	(x 1)2	,
	x2
M 0 ( 1;4).

+ ,. T y = f (x) M 0 (x0 , y0 ).

y y0 = f 8(x0 )(x x0 ). > f 8(x):

2(x 1)x2

2x(x 1)2

2(x 1)x 2(x 1)2

2x 2 2(x 1)

f (x) =

2( 1 1)

= 4,

( 1)3

f ( 1) =

.(-1,4) - :

y 4 = 4(x +1) y = 4x + 8.

= Oy . H , + Oy , x = 0.

x = 0, y = 8 . C, y = 4x + 8 Oy

(0,8).

H Ox y = 0. y = 0,

x = 2. = ", Ox ( 2, 0).

C , &

205

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