контрольная_2_ВведВ МатАнализЭЛЕКТР

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- 2006

. 2 34

. ., . .

! 1. xn :

1)2- , 100- , (n + 1)- xn ;

2), xn ;

3), !,

lim x

n

= A , > 0 N = N ( )

!,

n

&' ( ( n > N

xn A

< .

! 2. ) * & « » + !,

lim f (x) = A.

x x

, '- :

0,1

0,01

0,001

0,0001

…

! 3.	/ 0 !- .
! 4.	1 ! ! m 0 !- y = f (x) '
! xo ? 1 ( 0 ! 2 0 !- .
! 5.	/ ! 0 !- ,	3 3 !. 4
! xo ( 0 !-&		f !, ' +-

+ D f U{xo} ' ! xo .

! 6. 5 3 (- 0 ! 0 !- .

- 2006

. 3 34

. ., . .

1

1. x

n

=

2n2

+ 1

,

A =

2

, = 10 3 .

3n2 1

3

3.

3

+ 2

)

lim

x 6

;

x

3 +8

x 2

)

lim

x3 1

;

x)

x 1 sin (1

)

lim x2 ctg2 3x ;

x 0

1

sin x

+)

lim

a x

;

x a sin a

3

1

)

lim

ln x + 1

;

x 1

7

ln x + 1 1

log

4 x,

0 < x

1,

x = 1.

4. y =

+ x3 ,

x >

1,

m

o

6.

2x + 2

,

x > 2,

x + 3

)

y =

2 4 x2 , 2 x 1,

x 3

,

x > 1;

2.

lim

2x2 8x

+ 6

= 4.

x 3

x 3

5

x5 3

')

lim

5x + 2

;

x + 3 3x3+1 + 4 x3

4

()

lim

(x

);

x(x 1

x +

x2

x

3x 2

)

lim

;

2

x

x +

x

1

)

lim

e2 x2 + e 3x2 2

;

cos 4x 1

x 0

!)

lim

sin 3x

.

x 0

sin 8x

1

5.

y =

1 + 2x arctg

.

x4 1

')

y =

x3 + 2x2 + 3x

;

x

5 x

)

y = 4

9 x2

.

7. ) + ! ! & -

: q = 15 3 p , s = 1 + 4 p . E -

& - . F ( 0 ! ', - ( ! «! &*».

- 2006

. 4 34

. ., . .

2

1.

x

=

n2

, A =

1

, = 10 3 .

2

n

2n2 + 3

3.

2 3

)

lim

10 x

;

x 2

x 2

)

lim

16 x4

;

2)

x 2 sin (x

)

lim x3 ctg x3 ;

x 0

ln cos 2x

+)

;

(1 x)2

1

)

4

1 + ln

2 x

;

lim

x 1 3 ln2 x + 1 1

2x m2 ,

x 1,

4.

y =

+

x >

xo

= 1.

x,

1,

1

6.

x

,

x 0,

x + 4

)

y =

0 < x 2,

log2 x,

2x 3,

x > 2;

')

y =

x3

3x2

+

2x

;

x 2

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

2

+ 3

3

5

')

lim

x

x

x

;

x +

3x 3 7x

()

lim x

9x

2

+

1 3x ;

x +

x2

+ x

5

1 x2

)

lim

;

2

+ x +

3

x x

)

lim

72x

+ 53x 2

;

2x

arctg 3x

x 0

!)

lim

tg 3x

.

x tg 4x

5. y =

arcsin (x 2)

.

x3 4x2 + 3x

) y = 2tgx .

7./ ! ' !, ! ' ! ! ( -

7 % ( 3.

- 2006

. 5 34

. ., . .

3

1. xn = 1 + (0,1)n , A = 1, = 10 4 . 3.

)

lim

x + 1

;

x 1 6x2 + 3 + 3x

)

lim

16x

2

+ 3

x

4x ;

x +

)

lim

ln x 1

;

x e

x e

9

1

+) lim

x

;

7

x

1

x 1

)

lim

(sin 2x)tg 2 2 x ;

x 4

4x+1,

x 1,

4. y

=

xo

= 1.

x, x >

1,

m 3

6.

x (x + 3)

, x 0,

1

)

y =

4 x2

,

0 < x 2,

5 2x,

x > 2;

')

y =

log

2

xx 3

;

x

3

2. lim

2x2

+ 3x 2

= 5 .

x

1 2

x 1 2

2x2 + 3

5

')

lim

x

;

4x + 7

x

x

2 x + 3

3x2 1

()

lim

;

2

x

x 8

x

)	lim		ln cos m x		;
			arctg (x2 2)
	x 0
)	lim	earcsin 5x earcsin 2x				;
			3ctg (7 2 x)
	x 0
!)	lim ctg x ln cos 2x .
	x 0

5. y = 2 x . 16 x2

) y =		1 x	.
		+ e 1 x
1

7.q = 100 10 p + s = 100 + 10 p - -

p . / & - , ! - . 1-

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

- 2006

. 6 34

. ., . .

4

1.

x

n

=

3n 1

,

A =

3

, = 0,4 .

2. lim

10x2

+ 9x 7

=

5n + 1

5

x

+ 7 5

19 .

x 7 5

3.

)

lim

x 3

;

2x2 3x

9

x 3

)

9x

81x

2

+ 1

lim

;

x +

)

lim

1 cos 12x

;

x sin 3x

x 0

+)

lim

ln cos 5x

;

arcsin2

2,5 x

x 0

)

lim

e ax e bx

;

arctg x

x 0

5 x

2

x 2,

,

5

4. y

= 3

xo = 2 .

x > 2,

m + 3 x 2,

6.

x

,

x 0,

x + 1

)

y =

0 < x 4,

log2 x,

2

+ 14x 36,

x > 4;

x

')

lim

4x

3 x

5 + 2 3

;

+ 6x 3 x2

x +

1

x

()

lim

x2 (ln (2 + x2 ) 2ln x);

x +

cos 4x

;

)

lim

1 + xsin 3x

x 0

tg2 x

cos x

1 (a x)

)

lim

;

x a cos a

2

1

!)

lim (x 5)

1

cos

.

x 5

x

			6
	5. y =			x 1			.
		x3	+ 2x2		8x

') y =	2x2 x				;		x
						) y = 9 4 2x .
	x	2	2x

7. , !- y ( . '.) + &

y = 200 + 5x, ( x – ! - . 3 - !-

+ y = 185 + 8x . / ! ! ( - !

!- ' '?

- 2006

. 7 34

. ., . .

5

1.

x

n

=

2n + 3

, A =

2

, = 0,005.

3n

3

3.

;

)

lim

x + 2

3x 2

x 2

x2 x 2

)

lim

4x

2

+

x

3 2x ;

x +

)

lim

sin

(1 x)

;

x 1

x 1

+)

lim

tg ax sin ax

;

x 0

arcsin3 bx

5

1

)

lim

1 + x2

;

arctg6 3

x 0

x

4.

3x 4 ,

x 5,

xo

= 5.

y =

2 + m x + 8, x > 5,

x

6.

log

3

x

,

x 1,

)

y =

4x x2 3, 1 < x 3,

3 x,

x > 3;

2.

lim

3x2

5x 2

= 7 .

x 2

x 2

4x2

5

+ 2

')

lim

2x6 1

;

x +

x

+ 3 x4

7

2x3

3

4x3 5

()

lim

;

3

2x

+ 3

x

)

lim

tg 2x tg

4

x ;

x

4

)

lim

e8x 2

e4x 2

;

ln (1

+ 4x2 )

x 0

!)

lim (ctg ( x 4))sec (x 2).

x

5.

y =

4

x2 + 3x

.

x2 x 12

')

y =

x2

4x

;

x 2x2

)

y = 1 e1 (2 x).

7.) + ! ! & -

: q = 23 3 p , s = 5 + 6 p . 1 ! ! 3 3 p

0- ! ! 3 3 - & - M! ? N + ! + 2 3 ! '

3 - ?

- 2006

. 8 34

. ., . .

6

1. x

=

3n2 1

, A =

3

, = 10 3 .

2.

lim

4x2 14x + 6

= 10 .

n

4

4n2 + 1

x 3

x 3

3.

2x2 9x + 10

lim

5

3(x + 1)

;

)

lim

;

')

x

x 2 2x2 10x + 12

x + 4 x3 + 7x 19

3

3

x4

x3 + x

4x+3

2

2

+ x

)

lim

x

x

;

()

lim

;

4

x

x

+ 1

x

5x7 3x2

9

1

;

)

lim

;

)

lim

1 + 8x

x 0 sin 4x tg 3x

x 0

3 1 2x 1

1 (x 4)

ln (1 + sin 2 4x)

+)

lim

(ctg x)

;

)

lim

;

(e2 arctg x ex )2

x

4

x 0

sin 7 x

(1

)cos 2x .

)

lim

;

!)

lim

x

sin 8 x

x 2

x 1

ln x

4.	log3				x		,
	y =
	x2 mx + 16,
6.
		x 3,
			x
)	y =					,
		1 x
			2,

x 3,	x = 3 .	5.	y =	1 cos 4x	.
x > 3,
	o			4x2 x3

x 3,		')	y =		49		14x + x		2	;
3 < x
	2,				7	log	7 x	7
x > 2;
			y = 51 (2 4x).
		)

7.O ' 0 + 2 %. N ! ! -- ', 3 , 2,7

& ( e 2,7 ). E ! ( -3.

- 2006

. 9 34

. ., . .

7

1.

xn = 1 + (0.2)n , A = 1,

= 10 5 .

3.

3

2

)

lim

4x

;

x 2

2 + x

2x

)

lim

(2xtg x cos x);

x 2

)

lim

sin 2 x tg2 x

;

x 0

x4

+)

ln (9 2x

2 )

lim

;

sin 2 x

x 2

3

1

)

lim

1 + tg 2x

;

x 0

5

1 + arcsin 8x 1

3

2x,

x

1,

4.

y =

xo = 1.

x

m + 1,

x > 1,

6.

3

,

x 0,

x

)

y =

log

x,

0 < x 4,

4

2

+ 33,

x > 4;

12x

x

2.

lim

6x2 9x + 3

= 3.

x 1

x 1

x8 + 6

+

')

lim

x 6

;

x

+ 8 x8

+ 6 + 4

x16

6

2x2

+ 2x +

3

5x 2 7

()

lim

;

2

2x

+ 2x +

4

x

)

lim

arctg2 (2x 3)

;

x

2

3 2

e9 4x

1

)

m 4x 2 +5

lim

cos

;

x

x

!)

lim

2 sin 2 x 1

.

x

0 ln cos 3x

x

5. y = arctg x .

')	y =		x3	4x2 + 4x	;
				x 2

) y = e 1 x .

7.4 ' ! ( & 0 !- O !,

&* + 3 I

' x : ) '3: x =

'3 ( ! M):

1 ( 0 ! &* 3 0 !- .

I

I+ ; ')

x= (I ) .

I +

- 2006

. 10 34

. ., . .

8

1. x

= 2 10n 1 , A = 2, = 10 4 .

n

10n

3.

)

lim

(x2 + 2x + 1)2

;

x 1

x4 + 2x + 1

)

lim

x6 + 5

x8 3x3 + 5

;

4x4 5

x

3x2 5x

x2

+1

)

lim

;

2

3x

5x +

x

7

+)

lim

1 (x 1)

(x 1) a

1 ;

x

)

lim

arctg (x2 2x)

;

tg (3 x)

x 2

log2 x

,

0 < x < 2,

y =

x ! 2,

4.

x2 3x m,

xo

= 2.

6.

1 +

,

x 2,

2 x

x 4

)

y =

,

2 < x 4,

x 3

x > 4;

2x 8,

	2. lim	6x2 75x 39	=	81	.
		2x + 1		2
	x 0,5

')

lim

x2 x + 1 1

;

1 3

x 1

x

3

()

lim

1

x

3

x +

;

x

)

lim

e x ex

;

x 0 sin 8x sin 6x

)

lim (cos 2x)1 sin 2 8x ;

x

9

1

!)

lim

x 2

.

5

x 2 1

x 3

	1
5.	y =		.
		sin x

	') y =		9 3x		;
			x2 6x + 9

) y = 1 2 1(x 2).

7. O ' , ! ! & + ' ! -

! - (15 % ( 3), ' (	2300
?

- 2006

. 11 34