контрольная_2_ВведВ МатАнализЭЛЕКТР

. ., . .

9

1. xn = 2 + (0,3)n , A = 2, = 10 3 .

3.

3

)

lim

x2 + 3x + 8 2

;

x + x2 + 2x3

x 0

)

lim

sin 2 x tg2 x

;

5x4

x 0

2x2

+ x + 2

x2

)

lim

;

2

2x

+ x 1

x

5

+)

lim

1 + x2

1

;

ex 2

x 0 e2 x2

1

(3x b)

)

lim

cos b

;

x b 3 cos 3x

2.

lim 2x2 21x 11 = 23 .

x 11

x 11

4

x8 + 6

')

lim

x 6

;

x

+ 36

x

+

3 2x6 + 1

()

lim (x 1) ctg (1 x);

x

1

)

lim

ln cos 8x

;

x

ln cos 4x

)

lim

x

2

x

x

x ;

+

!)

lim

x

2

1 x

2

a

1 .

x

3mx 1,

x < 1,

2x 6x2

4.

y =

xo = 1.

5.

y =

.

5x x2 , 1

x 5,

(x 1)(1

16x2 )

6.

log x,

0 < x 2,

')

y =

(3 x)e

ln x

;

2

2x 2

3x 9

)

y =

, 2 < x 6,

4 x

x > 6;

1

x 1,

)

y =

.

1 + 21 (x 1)

7. ) + ! ! & -

: q = 23 3 p , s = 5 + 6 p . E -

& - . F ( 0 ! ', - ( ! «! &*».

- 2006

. 12 34

. ., . .

10

1.

x

=

2n2

5

, A

= 2, = 10 3 .

2.

lim

15x2

2x 1

= 8 .

n

n

2

+

7

x

+ 1 5

x 1 5

3.

3

1 2x + x2

1 x

x 5

27x

6 + x2

)

lim

;

')

lim

x

;

x2

x2

x 0

+ x

x +

+ 4 x

3

x sin 2x

2

3

2

)

lim

(x + 2)

(x 2)

;

()

lim

;

x

x 0 1 cos 5x

2

3x

+ 6

x 2

)

lim

x

;

)

lim

(aarcsin (5 x) 1)3 2x3 + 7 ;

2

x

+ 5x + 1

x

x

esin 3x esin 10x

1

(3x x 2 )

+)

lim

;

)

lim (cos (2x 6))

;

ln (1 +

x 0

3 x)

x 3

)

lim (2 x a)tg ( x 2a);

!)

lim sin 5x .

x a

x sin 8x

log

x 2

4x x2

,

0 x 2,

xo = 2 .

5. y =

.

4. y =

2

3mx2 2x,

x > 2,

x2 + 16x 4x3 4

6.

(3 2x) (x 1),

x < 4,

3

y =

2x

3

x

4

)

y =

12x x

2

32,

4

x 8,

')

;

x

0,5x 4,

x > 8;

)

y = 2x 2 ( x3 1) .

7.

q = 900 10 p +

s = 100 + 10 p

- p . / & - , ! - . 1-

( 0 ! 0 !- !	! + - p , !
! !;	2 ! & ! .

- 2006

. 13 34

. ., . .

11

1. x

=

n2 + 2

, A

=

1

,

= 10 3 .

n

3n2

3

1

3.

)

lim (1 + x)3 (1 + 3x) ;

x 0

x2 + x5

)

lim (

x) ;

x(x 2)

x +

)

lim

xsin 5x

;

cos3x

x 0 1

+)

lim sin 7x ;

x tg6x

)

lim xln

1 + x

;

x 1

x

x + x2 ,

x 1,

4. y =

x > 1,

xo = 1.

x 9,

m 3

6.

x + 1

,

x 0,

x + 2

0 < x 6,

)

x 1,

y =

+ 2,

x > 6;

x + 3

2.	lim	2x2	+ 15x + 7	= 13.
			x + 7
	x 7

')

4x2

4 x3

lim

;

x + 3 x6 + x3

+ 1 5x

x3

+ x

+ 1

2 x2

()

lim

;

3

x

+

2

x

)

lim

9ln2 (1 3x)

;

4arctg2x tgx

x 0

10

)

lim

1 + x3

1

;

arcsin3 x

x 0

cos 4a

a (4 x 4a)

!)

lim

.

x a cos 4x

5.	y =	arccos x
			.
		32x3 16x2 50x + 25

') y = 3x4 2x ; x

) y = 9(1+ x)(2x +8) .

7.O ' , ! ! ' 'P ! !- ,

3 ! ( + -

20 % ! 24 000 -

!- .

- 2006

. 14 34

. ., . .

12

1.

x

=

2n2 7

, A =

2

, = 0,005.

n

3n2 1

3

3.

)

lim (1 + x)3 (1 + 3x);

x 0

x2 + x5

x2

)

lim

x2

+ 3

;

x 4 x8 2 x

x2 6x

+ 5

4x 1

)

lim

;

2

5x

+ 5

x x

+)

lim

e2x 4 e2 x

;

x 2 ln(1 sin(x 2))

)

lim

sin (x 6)

;

x 6

3 2 cos x

mx2

+ 9,

x 1,

xo = 1.

4. y =

+ 5,

x >

1,

3x

6.

x 4,

x 4,

x + 4

)

y =

,

4 < x 0,

x +

2

x > 0;

log2 x,

	2. lim	6x2 x 1		= 5 .
		x 0,5
	x 0,5

')

lim

9 + 2x

5

;

x 8

3 x2 4

2

()

x

4

+ 3

x

4

lim x

2 ;

x

)

lim

n

x + 1

1

;

x 0 k

x + 1 1

)

1 tg 2

x

;

lim (cos x 2)

x 4

3x 1 1 (

1)

!)

x

.

lim

x 1

x + 1

5.y = cos ( x2).

x3 x2

')	y =		6 x2				;
			x2 6
			x		2
)				x		1
	y = e						.

7. q = 900 10 p + s = 100 + 10 p - p . 1 ! ! 3 3 p 0- ! ! 3

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

! ' 3 - ?

- 2006

. 15 34

. ., . .

13

1. xn = ( 1)n+1 , A = 0, = 10 2 . 5n

3.

)

lim

x4 1

;

x 1 2x4 x2 1

3

13x2

)

lim

x

;

x 9x 4 13x8 + 1

x2 3x +

6

x 2

)

lim

;

2

x

+ 5x +

1

x

+)

lim

ln (4x 3)

;

x 1 tg2 (5x 5)

)

lim

cos 0,5x

;

x esin x esin 4x

1 m

,

x 4,

y =

x

x > 4,

4.

0,5sin(x + 4),

xo = 4.

6.

2 3

,

x 1,

x

4 x

)

y =

, 1

< x 5,

x

4

x > 5;

x 3,

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

2

')

lim

x + 13

x + 1

;

9 x2

x 3

()

lim

x

2

+ 3x 2

x

2

3 ;

x

cos 2x (

1)

)

lim

cos 4x

;

(arcsin

)4

x +0

x

sin m cos 3 (x m)

) lim ; x m sin x

!)		lim	2x 3x	.
	x + 2x + 3x
5. y =		arcsin(x 4)			.
		x3 x2 20x

') y = x4 + x ; x2 + x

) y = 1 ex(3x+9) .

	7. 1	q =	2 p + 3	0 !- . / ' & !
			p + 0,5

0 !-& - . 1 ( 0 ! 2 3 0 !- .

- 2006

. 16 34

. ., . .

14

1. x

n

=

2n 1

, A = 1, = 10 2 .

2n

3.

)

lim

x3

6x2

+ 12x 8

;

x3 3x2

+ 4

x 2

14x3

)

lim

x5 + 1

;

x +

14x6

+ 3 x

5x2 + 3x 1

x2

)

lim

;

2

5x

+ 3x +

3

x

+)

lim

etg 3x ex

;

x 0 arcsin4 (

x

)

)

lim

ln cos 4x

;

x ln cos 6x

(9 x

2

),

3 x < 0,

y = m

4.

ex + 2,

x ! 0,

x0 = 0.

6.

	2.	lim	6x2 5x + 1		= 1.
			x 1 3
		x 1 3

5

')

lim

2x + 9

;

x 8

2 3 x

2

()

lim

x(x + 2)

x

2x + 3 ;

x

sin (sin x)) lim ( ) ;

x 1 ln ln x + 1

1 sin 2 x

)

lim

cos 3x

;

x 0

cos 4x

5

!)

lim

x 2

.

x 2

7 x 1 1

5. y =

arctg x

.

x3 3x2 4x

			0 < x 2,			x	2	x	3
	2log2 x,			') y =						;
								x
)		4 x,	2 < x 3,
	y =
	x 4

, x > 3; ) y = ex2 (x+5).

x 5

7.Q !- - ! 3 ( (

!: 3 0 Q1 p1 % , Q2 p2 %

. . , ! & 0 3 + ! 3. 1 ( 0 !.

- 2006

. 17 34

. ., . .

15

1. x

n

=

2n2 + 1

, A =

2

, = 10 2 .

5n2 + 3

5

3.

x2 + 3x + 2

)

lim

;

x 1 x3 + 4x2 +

3x

)

lim

3

x2 15

+ 3

x3 + 4

;

3 15x3 1

x + 4

x

;

)

lim

1 + tg x

1 + sin x

x3

x 0

+)

lim

ln cos 10x

;

x ln cos 6x

)

lim

(tg x)ctg x ;

x 4

2

,

5 x < 2,

2m 5 x

y =

3

x + 1,

x ! 2,

4.

2

xo

= 2.

6.

3

,

x 0,

1 x

)

y =

log1 3 x

,

0 < x 3,

x > 3;

x2 8,

2.

lim

3x2 2x 1

= 4 .

x + 1 3

x 1 3

2

4

')

lim

x

;

x 16

x

4

2

()

lim

x(x 2)

x

3

;

x +

x2

+ 2x + 1

2x2

)

lim

;

2

+ 2x

3

x x

(e x + 1)(arcsin

)2

)

lim

x

;

x +0

3 1 + x2

1

!) lim 2x 7 1(3 x 2).

x 8 x + 1

5. y = 4x3 4x2 + x . 1 2x

')	y =	2log2 x 1			;
			x2	2x + 1

) y = 1 e 1x2 .

7.Q !- ! U B ' :

U = 18B 0,5B2 , ( B – ! ' ( - -

. / 0 !-& . 1 ( 0 ! - ! . R ! !

' 10 -? 1 ' 15 -? )! ! -

B + ', ! ( ?

- 2006

. 18 34

. ., . .

16

1. x

=

3n2 + 2

, A =

3

, = 10 3 .

n

4n2 1

4

3.

(x2 + 2x 3)2

)

lim

;

x 3

x3 + 4x2 + 3x

4

x4

+ 2 +

)

lim

x 16

;

x + 4 x4

+ 2

x 16

3x2 7x + 1

x2 5

)

lim

;

2

3x

7x 2

x

3x 2

e

1 arccos 3x

+)

lim

;

ln

(1 + sin 2 x)

x 0

)

lim

2sin x 1

;

6x + 10)

x 3 ln (x2

3

,

x 1,

x

y = 1

(4x 2), x > 1,

4.

m

6.

xo

= 1.

x

2

,

x 0,

)

y =

log

x,

0 < x 4,

2

2 + 3

x > 4;

x 4,

2.

lim

6x2 + x 1

= 5.

x 1 3

x 1 3

3

+ 2

')

x 6

;

lim

x3 + 8

x 2

2

3

()

lim x

5 +

8x

3

2x ;

x

1 8

)

lim

x 1

;

x 2

9 x

1 1

)

lim

4

(tg x) tg 2x ;

x

!)

lim

ln cos 10x

.

x ln cos 4x

5. y =

x + 1

.

arctg (1 x)

')

y =

x2 3x

;

x + x2

2

x

+1 x

)

y = e

.

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

p . / & - , ! - . 1

( 0 ! 0 !- !	! + - p , ! !
!;	2 ! & ! .

- 2006

. 19 34

. ., . .

17

1.

x

n

=

n2

, A =

1

,

= 10 3 .

2.

lim

x2

+ 5x + 4

= 3 .

4n2 1

4

x + 1

x 1

3.

)

lim

x3 3x 2

;

(x2 x 2)2

x 1

4x2 + 15x + 17

3x+9

)

lim

;

2

4x

+

5x + 10

x

)

lim

tg x

4x3

;

x 0 arcsin 3

8x3

1

2

x

2

sin x2

b

+)

lim

;

2

x b sin b

)

lim

sin 5x

;

x tg 3x

y =

2

,

2 x 2,

4 x

4.

mx + 2,

x > 2,

xo

= 2.

6.

2x + 4

,

x 0,

2x + 2

) y =

1 4 + 2x x2 , 0 < x 2,

(x 3)2 ,

2 +

x > 2;

')

lim

1 + x

1 x

;

x 0

5 x

()

x

2

3x + 2

lim

x ;

x +

)

lim

ln (1 2x)

;

x 0 e 3x e 5x

)

lim (4 x)x

(x2 9);

x 3

1 10

!)

lim

x 1

.

x 2

12 x 1 1

5.y = lg1x .

')	y =		x2	10x + 25	;
				5 x

) y = 61(2x 4) .

7.1 !, + ' ! 15 % ( 3,

4 . '. / ! ! ! 5 - - : ) + (; ') !; ) ?

- 2006

. 20 34

. ., . .

18

2n2 1

2

2

1. xn =

, A =

,

= 10

.

3n2

3

3.

)

lim

(x3 2x 1)(x + 1)

;

x4 + 4x2

5

x 1

6x2

)

lim

x5 + 1

;

x +

4x6 + 3 x

2x2

+ 18x + 1

2x 3

)

lim

;

2

2x

+16x 1

x

+)

lim

ln(1 + sin(x 2))

;

x 0

e4 arcsin x 1

)

lim

sin 7x

;

x tg 5x

log3 x,

0 < x 3,

y =

> 3,

x

3,

x

4.

m

xo

= 3.

6.

2.

lim

x2

5x

+ 4

= 3 .

x 1

x 1

5

;

') lim

9 + 2x

x 8

3 x 2

()

lim

(x

);

x(x 1)

x +

)

lim

1 cos12x

;

x 0

x tg 5x

)

lim (sin x)tg x ;

x

2

7

1

!)

lim

x

.

x 1

9 x 1

y =

log2 (2x + 1)

5.

.

x3 10x2 4x + 40

			3
															2
				x 1,			x 2,
	1							')	y =			(x 1)				x2 ;
)	y =	9 2x			,		2 < x	5,					x 1
		4 x
							x > 5;
		x 4,								1				2
								)	y =				3x		12
										5						.

7.) + ! ! & -

: q = 19 2 p , s = 3 + 2 p . E -

& - . F ( 0 ! ', - ( ! «! &*».

- 2006

. 21 34