контрольная_2_ВведВ МатАнализЭЛЕКТР

. ., . .

19

1 + 3 10n

3

2

1. xn =

, A =

, = 10

.

5 10n

5

3.

)

lim

x3 4x2 3x + 18

;

x2

6x + 9

x 3

)

lim

1 + x

2x

;

x 1

1 3 x

3 x +

5x2 10 4x

)

lim

;

2

5x

x

+ x 1

+)

lim

cos 4x cos 2x

;

x 0 e 2 arcsin 2 x esin 2

cos x

1 (x 5)

)

lim

;

x 5 cos 5

1 cos (x 1),

x 1,

y =

m

+ 3 x 1,

x > 1,

4.

xo

= 1.

6.

,

x 0,

4 x

)

x,

0 < x 8,

log

1 2

x > 8;

8 4,

x

2.

lim

x2 8x + 7

= 6 .

x 7

x 7

+ 5

x

6

2x10 + 1

')

lim

x

;

(x + 4

)4 x7

x +

x

()

lim

x

2

3x + 2

x

;

x +

)

lim

ln (17 x

2 )

;

tg ( x 4)

x 4

)

lim

ln cos 2x

;

x ln cos 4x

!)

lim

arctg2

(2 x)

.

x 2 5 x2 4x + 5 1

5. y =

16 x2

.

24 + 4x 6x2 x3

') y = 2log2 x 4 ;

4 x

) y = 1 e2(2+ x) .

7.4 ' ! a % ( 3 ' + ! . N n

! b . '. R ! + ?

- 2006

. 22 34

. ., . .

20

1. xn

3.

)

)

)

+)

)

=

1 2n2

, A =

1

, = 10 3 .

4n2 + 3

2

lim

x3 3x 2

;

(x2 x 2)2

x 1

2

;

lim

x + 13

x + 1

9 x2

x 3

4x2 + 3x 7

11 x

lim

;

2

4x

x + 1

x

lim

sin (1 2x) tg x ;

x 0,5

arcsin2

lim

3x

;

x +0 ln (1 + 12tg x)

2.

lim

x2 8x 9

= 3 .

x 2

x 9

3

5

3x6 x2 + 1

')

lim

x

;

2x2 x +

10

x

1

1

()

lim

;

2

x 1

x

x 1

1

)

lim

ln cos 2x

;

(1 x)2

x

6

)

x2 4x + 5

1

;

lim

e4 x 2

x 2

1

sin x

1 (x 2)

!)

lim

.

x 2 sin 2

log

5 x,

0 < x 5,

y =

x > 5,

5. y =

1

4.

m

x 5,

.

ln

x 1

xo

= 5.

6.

(x + 3) (x + 5),

x 1,

')

y =

4x + x2

;

2x2

x

)

y =

3 x + 2,

1 < x

7,

x 5,

x > 7;

1

)

y =

.

1 e

x(1 x)

7.4 ' ! ( & 0 !- O !,

&* + y 3 x

	': ) ! M					y =	a(x b)		, x ! b ;
							x + c
			ax(x b)
	') ! M y	=			, x ! b , ( a,	b c –			+-
			x + c

. 1 ( 0 ! &* 3 0 !- .

- 2006

. 23 34

. ., . .

21

1. x

= 2n2 1

;

A

=

2 , = 10 3 .

3.

n

3n2 + 1

3

)

lim

x4 1

;

2x4 x2

x 1

1

2

x2

+

6

)

lim

x 6

;

x + 7 x

7

2 +

x

1

3x2

4x

2 x 13x 2

)

lim

;

3x

2

4x + 7

x

+)

lim

3sin 6x

;

x tg 4x

)

lim

ln cos 6x

;

x ln cos 4x

3x+ 2 ,

x 2,

y =

x2 ,

x > 2,

4.

m

xo

= 2.

6.

2.

lim

3x2 + 17x 6

= 19 .

x 1 3

x 1 3

5

;

')

lim

9 + 2x

x 8

3 x2 4

1

1

()

lim

;

x 1

x 1

4 x 1

ln (1 + arcsin2

)

)

lim

x

;

earctg x 1

x +0

6

1

)

lim

1 + 12x2

;

x +0

cos 2 x 1

x

1/(x 2)

!)

lim

tg

.

2

x 2

5. y = arcsin x . sin 2x

			log2		x		,
	)		4x x2 3,
		y =
			2 x,

x1,

1< x 3,

x> 3;

') y = x3 2x2 + 3x ; x

) y = e x (6 2x).

7.1 M -3, 0 ' !!. ) – 300 . . . 1 +

– 3 (. ) ! – 10 % (. T ! (

0: 1) – 100 . . . (; 2) &* – ( 60, 100 140 .

. . R ! 0 (?

- 2006

. 24 34

. ., . .

22

1. x

=

2n + 3

,

A =

2

, = 10 2 .

n

3n

3

3.

)

lim

x2

+ 2x 3

;

x 3 x3 + 4x2

+ 3x

x2

+ 2

5x2 +

)

lim

x

;

x +

x4

x + 1

4x2

+ 3x

1

17 x2

)

;

2

lim

4x

+ 3x + 5

x

+)

lim

ln (5 x2 )

;

1

x 2 cos 4 x

)

lim

ln cos 4x

;

x 2 (1 2 x)2

4 3x,

x 1,

y =

x > 1,

x

m

1,

4.

xo = 1.

6.

2.

lim 2x2 + 3x 2

= 5.

x 0,5

x 0,5

3

;

')

lim

1 + x

x 8

2 3 x

()

lim

(

);

x + 3

x

x 4

x +

)

lim

e tg 2 6x 1

;

x 0 cos 9x cos 3x

)

lim

arcsin2 (1 x)

;

x 1

7 x2 2x + 2 1

!)

lim (2 x 3)1 sin x .

x 3

5. y =

arctg x

.

x3 + x2 2x

3

2

x ,

x 0,

x

x

')

y =

;

x2 + 2x

)

y = log2 x,

0 < x

2,

(x 3)2 ,

x > 2;

y =

1

)

.

1 x 1

1 + e

7. q = 600 8 p + s = 120 + 8 p -

p . 1 ! ! 3 3 p 0- ! ! 3 -

3 - & M! ? N + ! + 2 3 -

! ' 3 - ?

- 2006

. 25 34

. ., . .

23

1. x

=

2n + 1

,

A =

2

,

= 0,05 .

n

3

3n 1

3.

3

+ 2

)

lim

x 6

;

x2 4

x 2

)

3

(x

2

3

(x

2

lim

+ 1)

1)

;

x

)

lim

8x3

5x2

;

x +0

2(1 cos 4x) arctg 2x

+)

3

x2 2x + 2

1

;

lim

arcsin (1 x)

x 1

)

lim

ln (2 + cos x)

;

x

(3 sin x 1)2

1 m

,

x 1,

x

y =

4.

x2 3x + 1, x > 1,

xo = 1.

6.

2.

lim

4x2 14x + 6

= 10 .

x 3

x 3

x 5

2

;

')

lim

x15 + 1

x

7x4 + 4

x2

3x + 6

x 2

()

lim

;

2

+ 2x + 5

x x

)

lim (2 x) sin ( x 2)

ln (2 x);

x 1

)

lim

tg2 ( x)

;

ln (10 x2 )

x 3

!)

lim

esin x 1

.

sin 4x sin 6x

x 0

5. y = x2 3x + 2 arctg (1 x).

				3
						x ,			x 1,					2
	2									')	y =		(x 2)				+ x	2 ;
		3	x						< x 6,					x 2
)	y =						, 1
		x			2
									x > 6;
			3 4,							)	y =		1 (5x +10)		.
												2

7.O ' ! -! + 2! -

: 2! 20 % ; 2! 30 % ;

2! 15 % . )! ! 2! - +?

- 2006

. 26 34

. ., . .

24

1.

1

,

3

,

5

, K , A =

2

, = 10 3 .

4

7

10

3

3.

)

lim

x3

+ 2x2 x 2

;

2x2 x 1

x 1

3

24x3 + x

)

lim

3x 1

;

x + 4 4x4 + 1 3 x2

1

3

)

lim

x

2

+ 2

3

x

2

3 ;

x

+)

lim

ln (cos 4x)

;

x 0

ln (cos 12x)

)

2x 1

ctg (1 x)

lim

;

x

x 1

m x3 ,

x 2,

4.

y =

x > 2,

2log4 x,

xo = 2.

6.

(1 x) (x + 2),

x 0 ,

)

y =

x + 0,5,

0 < x 2,5 ,

x 2,5 + 3,

x > 2,5 ;

2. lim	2x2 x 3 = 1.
x 2		x + 1
	4				1
			x
') lim							;
x 1 1 x2
()	lim				3x		2x	;
							+ 2x 1
x + 3x 1

arcsin3 2x

) lim ( ) ; x 0 arctg x 1 cos x

)

lim

5 x2

6x +

10 1

;

sin x

x 3

!)

lim (cos x)1 tg (x 2) .

x 2

5.

y =

x3

x2

12x

.

x

arcsin

4

')	y =	1 x 1 (x + 1)		;
		1 (x 1) 1 x
)	y = 5x (1 x2 ).

7. N (: ' ' 5 . . .. ! ' ( 5 ) 7,5 . . . .? 1-

! 10 % ( 3.

- 2006

. 27 34

. ., . .

25

1. x

=

1 + 3n

, A = 3, = 10 3 .

n

6 n

3.

5

;

)

lim

9 + 2x

x 2

3 x3 8

2

)

lim

9 + 2x3

x

;

x +

3

x6

8

5x3

2

6x3

)

lim

;

3

x

5x

+ 2

+)

lim

ln(3 x) ln 3

;

x 0

5 x

)

lim (1 x)ln (1+ x ) ln (2 x ) ;

x 1

x3 8

x " 2,

y = x

2 ,

.

4.

m,

x = 2,

xo = 2.

6.

sin x

x < 0,

)

,

x

y =

x + 1,

x ! 0;

2. lim x + 6 = 4 .

x 2 x

')

lim (

x);

x(x + 5)

x +

3

()

lim

10 x

;

sin x

x 1

)

lim 3 x2 ctg2 2x;

x 0

)

lim

ln(2x 5)

;

x 3

esin x 1

!)

lim

(1 + 3x2 )5

1

.

5 x2

x 0

5.y = arcsin (4x) .

x2 4x

')

y =

x 1

x

;

x

x + 1

)

y =

4

.

3 + 5

3 (x 2)

7.1 + * 0 0 - + -

& : 10 . . . 2 . . .

5 , ' 15 . . . 1 . . .

M . 1- ! (! 200- !) 10 % ( 3. R ! (?

- 2006

. 28 34

. ., . .

26

1 + 5n

3

1. xn =

, A = 5, = 10

.

6 n

3.

4

;

)

lim

8 + x

x 8

3 x 2

)

lim

9 + x

4

+ x2

;

x

3 x6

8

)

lim 3x ctg 4x;

x 0

+)

lim

ln x ln 4

;

x 4

x 4

)

lim

(1 + 2

x3 )x3

;

x

ex

1

,

x " 0,

x

y =

4.

m,

x = 0;

xo

= 0.

6.

x

,

x

> 1,

)

y =

2

,

1 x 1;

x

2. lim

x2 1

= 2 .

x 1 x 1

')

lim

x

2

+

5x x ;

x +

()

lim

3

9 x

2

;

1 cos

x

x 0

3x2

2

x2

)

lim

;

2

3x

+ 2

x

)	lim		ln(2 + cos x)			;
			(3 sin x 1)2
	x
!)	lim		1 cos x	.
	x 0	(e 2 x 1)2

5.y = 2x(1 xx) 1.

') y = arctg (1x);

) y = 4 x2 . x 2

7.U ! 0 , ' -

' 10 000 . R ! + ' +

+ ( ! ! ), - ! -

12 % ( 3?

- 2006

. 29 34

. ., . .
						27
1.	xn	=	1 + 5n		, A = 0, = 10 2.	2. lim	3x2 + 1		=	3	.
				n2
										2
						x 1 2x2
3.

)

lim

1 + x x2

1

;

x 0

3 x + 8

2

)

lim

(3 x)4 (2 x)4

;

x (1 x)3 (1 + x)3

)

lim 2x ctg 5x;

x 0

+)

lim

ln(x 2)

;

x 3

2x 8

)

lim (2 cos x)1 x 2 ;

x 0

ex 1

,

x " 0,

x

y =

4.

m,

x = 0;

xo = 0.

6.

)

y =

x2

,

x

> 1,

x

,

1 x 1;

2

3

')

lim

x

x

3

+ 5

x ;

x +

()

lim

x3 + x2

;

1 cos 2x

x 0

x2

2x 2

x

)

lim

;

2

x

+ 2x + 2

x

)

lim

ln (1 2 x)

;

x 0

5arctg3x

!)

lim tg x

cos(x + 3 2).

x 0

(arcsin x)2

5.y = 2x(1 xx) 1.

') y = arctg (1x );

) y = (x 2)2 (4 x).

x 2

7.4, ! ! & ' ! 10 % ( 3,

' 5 10000 '?

- 2006

. 30 34

. ., . .

28

1. xn

=

5

, A = 0, = 10 2.

5n2

+ 3n

3.

x2

)

lim

x

;

x 1

x

1

)

lim

(3 x)3 (2 x)3

;

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

)

lim x ctg (5x + x2 );

x 0

+)

ln(1 + tg

2 x)

lim

;

ln cos x

x 0

)

lim (2 x)2 (x 1);

x 1

2x

1

sin x ,

x " 0,

y =

4.

m,

x = 0;

xo = 0.

6.

x2 1,

x 1,

)

y =

1

e

1 x

,

x > 1;

2. lim	5x + 1		=	5	.
				2
x 1 2x

')

lim

x

2

+ 4

x ;

x +

()

lim

tg 5x

;

1 cos

x +0

2x

)

5x 2

4x

lim

;

x

5x + 2

)

lim

ln (1 + 3 x)

;

x 0

earctg 3x 1

2x3 + 1

!)

lim

5x

.

x +

2x (sin x + x)

5.y = sgn(x2 1).

') y = arctg (1(x 2));

) y = ln x .

7. R ! & ' ! 15 % ( 3, ' + (

&' ( ! + ' ! 2000 . .?

- 2006

. 31 34