attachments_22-12-2011_21-07-09 / razdel 4

1

. 7 n > N

0

1

< , !

n

8, xn = n2 .

. xn

&,

& M ( (

N ,

n > N |

xn |> M .

H ! $ :

lim xn

=

xn

n

.

n

. xn

+ , &

M

(

(

N ,

n > N

xn

> M.

:& ! :

lim xn = +

xn

n

+ .

n

. xn

, &

M

(

(

N ,

n > N

xn

< M .

:& ! :

lim xn =

xn

n

.

n

R ( ( () -

R (x0 ) = { x :

x x0

< } R (x0 ) = (x0 , x0 + ) .

5 R (x0 ). - ! 2 ( x0 , &

( . 4.2a).

)

R ( x0 )

&)

R ( x0 )

R+ ( x0 )

x0

x0

x0 +

x0

x0

x0 +

4.2.I

N & ( , "

" , ! !.

x0 - .

H R (x0 )

-

x0 R+ (x0 )

-

x0 .

, R (x0 ) = (x0 , x0 ) ,

R+ (x0 ) = (x0 , x0 ) ( .4.2&). 8

-

.

. X (

x0 &

R (+ ) =

" 1

, +

#; R ( ) =

"

,

1 #

;

R ( ) = ( ,

1

) (

1

, + ),

%

%

&

&

'

(

'

(

4.3.I

M,

x R (+ ) , !

x >

1

,

x R ( ) , !

1

1

x <

, x R ( ) , ! | x |>

( .4.3).

( x a , a

- ) f (x) R ( A) ,

A

f (x)

a ( x , " a ).

H ! $ lim f (x) = A f (x)

A x

a .

x a

H , & a

A ,

1) lim x = a ,

x a

const , lim f (x) = lim C = C.

2) f (x) = C ,

x

a

x a

M ($ ( .

?. # x

a f (x)

,

! .

?. # x

a

f (x)

,

?. # x a f (x)

A ( X

a !

( ), A 0 ( A 0 ).

f (a 0),

lim

f (x),

lim f (x).

x a 0

x

a

x<a

. # &

) = )( ) , "

,

x R+) (a)

f (x) R ( A) ,

A

f (x) a &

f (a + 0),

lim

f (x),

lim f (x).

x a+0

x a

x>a

N (

(

a

! ( . A .

4.4 ,

? (

! ).

f1(x), f2 (x),

f3 (x)

( a ( ,

&,

! , a - ), ! (

f1 (x) f2 (x) f3 (x).

? (o + ! ).

f (x)

a . 7

" ( (

f (x)

a ,

a -

,

(

f (x)

x

a ,

a = +

a = .

A

! ,

,

"

(

"

+

1

#x

lim %1

& . E - , & & ( e

x

x '

(

( x

a ),

lim +(x) = 0 .

x a

. @ f (x) ! ' a

( x

a ),

lim f (x) = .

x a

? ( ! ! '). #

+(x) - & a ,

1

- & &$

+(x)

1

! ( ; f (x)

- & &$

a ,

- &

f (x)

" 1

+

2

#

%. H

lim %

& .

x

2

x ' x

(

1

& '. 7 x

,

&

- & .

x

1

2

"

1

2

#

M &

- & . >, lim %

+

& = 0 .

x

x

2

x

2

5

x

' x

(

%. H lim x2 sin

.

x

0

x

& '. x2 - & ( &

x , x ), sin

5

- . 7, ( 2

&

x

5

5

(,

x2 sin

- & , . . lim x2 sin

= 0 .

x

x 0

x

# ! '

1. # x a

g(x)

,

f (x) - &

&$

x

a ,

g(x) , f (x)

& &$( ( x

a.

2. #

lim f (x) = +

lim g(x) = +

,

lim - f

( x)

+ g ( x). = +

.

x a

x

a

x a /

0

#

lim f (x) =

lim g(x) =

,

lim - f

( x)

+ g ( x). =

.

x a

x

a

x a /

0

3. #

f (x) - & &$ a ,

g(x) -

a ,

f (x) + g(x)

- & &$ a .

M & &$ (, " ,

& & &$(.

3

2

"

+

#

%. H lim%

& .

2

x

4

x

0 ' x

(

1

& '. x

0 x2

x4

- & . 7

x2

1

- & &$ .

lim 3 = 3 0

lim 2 = 2 0 ,

,

3

x4

x2

x

0

x 0

2

- & &$ (( 1), , & . 7

x4

( 2

+

lim

"

3

+

2

# = +

.

, . . x 0

'% x2

x4 (&

?.

C

&

f ( x)

x

a

A ,

&

,

&

f ( x) A = ( x)

& & ( a .

M " ! ( !

x a

f1(x)

f2 (x)

?.

#

2.

lim[ f1(x) + f

2

(x)] = lim f1

(x) + lim f

2 (x);

x a

x a

x a

lim(3x2 + 5x 8) = 3(lim x)2 + 5lim x lim8 = 3 , 22 + 5 , 2 8 = 14 .

x 2

x 2

x

2

x

2

& ' !

% 1. ,

{x } =

sin n

.

n

n + 2

& '.

, |

xn |

& n

. C (, ,

. >,

| sin n | 1 .

1 . . n 1 ,

n + 2 > 1 + 2 = 3 .

7

sin n

1

n ,

n + 2

3

.

{xn} = {2 + 3n}

% 2. ,

.

. 7 n 1 , xn = 2 + 3n 2 + 3 ,1 = 5 ,

& '. : xn

.

% 3. ,

{xn} = {5 n}

.

: xn

. 7 n 1 ,

xn

= 5 n 5 1 = 4 .

& '.

E

, .

{xn} = {( 1)n n}

%

4. ,

, .

xn |= n .

& '.

> , |

L & M > 0

, ( , &$ ! M . 7. . & M > 0

(

x

| x |> M

,

{

| x

|

n ,

n

n

}

&$ & &$ ! M .

7. . (

xn , $ (- M ).

E ,

{xn} , .

1

%

5. ,

{xn} =

n -

3

n

& ".

& '. , ( " (

$ ". I

x

x =

" 1

(n +1)# "

1

n

#

=

1

n 1

1

+ n =

1 " 1

1#

1 =

2

1 < 0

,

%

&

%

n+1

n

n+1

& %

n

n+1

n

n

&

n+1

(

' 3

( ' 3

( 3

3

3 ' 3

3

&, lim x cos(2x 8) = 0 .

x

0

1

"

1

#

% 7. H

lim

%5 +

& .

2

3 + x

2

x

x

'

(

& '.

@

1

,

&

& &$(, - &

x2

. , %"5

+

1

&#

-

. C (,

3 + x

2

'

(

x2

0 ,

1

0 , , , 5 +

1

5 . M ( ,

3 + x2 3 ,

3

+ x2

3 + x2

, ,

1

1

.

7

5 +

1

5 +

1

= 5

1

. 7 &,

3 + x2

3 + x2

3

1

3

3

,

5 +

,

, ,

3

+ x

2

.

& (

–

& . ! lim

1

"

5

+

1

#

= 0 .

%

&

2

3 + x

2

x

0 x

'

(

% 8. H

lim

(x2 + sin 2x) .

x

& '.

x

x2

-

& &$ ,

& &$ ,

sin 2x

-

, –

& &$ . M,

lim (x2

+ sin 2x) = .

x

"

2

7x +

2

#

% 9. H lim%5x

& .

x

x

1

'

(

& '. C $ ! (

, ,

lim x = a

"

2

2

#

(lim x)

x

a

2

2

2

lim%

5x

7x +

& = 5

7 lim x +

= 5 ,1 7 ,1 +

= 0 .

x

lim x

1

x 1 '

(

x

1

x

1

3

x

1

% 10. H lim

.

x

2

x

2

& '. I

lim ( x 2)

(

x 2

.

7

lim ( x 2) = lim x lim 2 = 2 2 = 0 . 7 &,

x

2