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### Transcript of Κατηγορίες ασκήσεων στα όρια

• N

f :

x0 = - 2, - 1, 1

-

-

.

-

- +

+

-

+

0

x - 2

x - 2 x - 2

x - 2

0

0

x - 1

x - 1

x = - 2

lim f(x) = 2 lim f(x) lim f(x)

lim f(x) = 1

f x = - 2.

x = - 1

lim f(x) = - 1

lim f(x) = - 1

- +

-

- +

+

x - 1 x - 1

0 x - 1

0

x 1

x 1 x 1

x 1

lim f(x) = lim f(x) = - 1

f x = - 1 lim f(x) = - 1.

x = 1

lim f(x) = 3 lim f(x) lim f(x)

lim f(x) = 1

0 f x = 1.

:

-

f .

:

-

f, C f .

:

-

.

:

1. x 0

.

2. - 0x x

lim f(x)

= + 0x x

lim f(x)

=

0x x

lim f(x) =

.

:

x 0

, :

:

-

C f .

:

-

C f .

002

y

3

2

1

-2 -1 0 1 x

• x - 1

, :

lim(3 - 2x) = 5 .

, > 0 -

> 0, :

|f(x) - 5|< |3 - 2x - 5|< |-2x - 2|< 2|x + 1|<

|x + 1|< .

2

x 0 0 -

> 0, :

x 0 < |x - x | <

|f(x) - |< .

:

0x x

lim f(x)

= .

1. | f ( x ) | < ,

| x x | < .

2. = / ,

| x x | <

|f(x) - | < .

003

• 2

x 1

2

x 0

x

x 0

2

x 2

N :

lim(3 - 2x + x )

lim(2 x + x)

limln(1 + e - e )

x + 5lim

2x - 1

-

2

0

2 2

x - 1

2

0

2 2

x 0

x = - 1, f(x) = 3 - 2x + x

lim(3 - 2x + x ) = 3 - 2(- 1) + (- 1) = 3 + 2 + 1 = 6

x = 0, g(x) = 2 x + x

lim(2 x + x) = 2 0 + 0 = 2 1+

x

0

x 0

x 0

2

0

2 2

x 2

0 = 2

x = 0, h(x) = 1+ e - e

lim ln(1+ e - e ) = ln(1 + e - e ) = ln(1 + e - 1) = lne = 1

x + 5 x = 2, r(x) =

2x - 1

x + 5 2 + 5 lim =

2x - 1 2 2 - 1

4 + 5 9 3= = = = 1

4 - 1 3 3

:

.

:

-

.

:

-

-

x0 .

:

1. x = x0

f(x) .

2. E

.

004

• 2 2

2 2x 1 x 1

N ,

:

x - x - lim = lim 1 .

x - x -

2 2 2

2 2 2 2x 1

2

x 1

2 2

x - 1- lim =

1- 1- x - 1- =1- 1- x - 1-

lim =x - 1-

(1- )(1- ) = (1- )(1- )

(1- )(1- () 1

+ ) - (1+ ) = 0

(1- )(1- )(1+ - 1- ) = 0

(1- )(1- )( - ) = 0

= 1

(1- )(1-

)

= 1

=

= = 1

:

.

:

.

:

-

-

.

:

1. -

.

2. -

.

3.

.

005

• o o

0

x x x x

N f g x , :

lim(3f(x) - g(x)) = 3 lim(2f(x) + 5g(x)) = 19

0 0x x x x

A h(x) = 3f(x) - g(x) p(x) = 2f(x) + 5g(x)

lim h(x) = 3 lim p(x) = 19 (1)

h(x) = 3f(x) - g(x) 5h(x) = 15f(x) - 5g(x)

p(x) = 2f(x) + 5g(x) p(x) = 2f(x) + 5g(x)

5h(x) + p(x) = 17f(x)

p(x

0 0x x x x

5h(x) + p(x)f(x) =

17) = 2f(x) + 5g(x) 10h(x) + 2p(x)

p(x) = + 5g(x)17

5h(x) + p(x)5h(x) + p(x) f(x) =

f(x) = 1717

3p(x) - 2h(x)17p(x) = 10h(x) + 2p(x) + 85g(x) g(x) =

17

5h(x) + lim f(x) = lim

0 0

0 0

0 0

(1)x x x x

(1)x x x x

x x x x

5 lim h(x) + lim p(x)p(x) 5 3 + 19= = = 2

17 17 17

3 lim p(x) - 2 lim h(x)3p(x) - 2h(x) lim g(x) = lim = =

17 173 19 - 2 3

= = 317

:

() -

-

f(x), g(x) .

:

E :

0 0x x x x

lim f(x) lim g(x)

.

:

f(x),

g(x)

.

:

1. h1(x), h2(x) -

-

.

:

0 0

1 2x x x xlim h (x) lim h (x)

.

2.

f(x), g(x)

( h1(x),

h2(x)) .

3.

0 0x x x x

lim f(x) lim g(x)

0 0

1 2x x x xlim h (x) lim h (x)

.

006

• 3

2x 3

N :

x - 27lim

x - 9

3 3

2 2

3 3

2 2x 3

2

x 3

(

x = 3 :

x - 27 3 - 27 27 - 27 0 = = = ,

9 - 9 0x - 9 3 - 9

x - 3 = lim =

x - 3

(x + 3

(x - 3))

= l(x

im3

- )

3

2x 3

x - 27lim

x - 9

2

x 3

(x -

x + 9) =

(x + 3)

x + 3x + 9 = lim =

x + 3

=

3)

9

2

:

(), -

x 0 .

:

E .

:

-

.

:

1. -

(-

Horner,

x0) .

2.

x - x0 .

3.

.

007

• x 0

N :

x + 4 - 2lim

x

/

x 0x+4 +2

x

( x)

x = 0 :

x + 4 - 2 0 + 4 - 2 2 - 2 0 = = = ,

x 0 0 0

( x + 4 - 2)( x + 4 + 2)= lim =

x( x + 4 + 2)

= lim

x 0

x + 4 - 2 lim

x

2 2

0

x 0

x 0

( x + 4) - 2 =

x( x + 4 + 2)

x + 4 - 4 = lim =

x( x + 4 + 2)

limx

=

x=

( x + 4 + 2)

1 = =

4 + 2

=1

4

:

-

.

:

E .

:

-

.

:

1. -

(

-

) .

2.

x - x0 .

3.

.

008

• x 1

x - 1N : lim

3 x + x + 3 - 5

x - 1 1- 1 0 0 x = 1 : = = = ,

3 + 2 - 5 03 x + x + 3 - 5 3 1 + 1+ 3 - 5

.

3 x + x + 3 - 5 (3 x - 3) + ( x + 3 - 2) 3 x - 3 x + 3 - 2= = +

x - 1 x - 1 x - 1 x - 1

x 1 x 1

x 1

3 x - 3 x + 3 - 2 .

x - 1 x - 1

3( x - 1) 3( x - 1)( x + 1) = lim = lim =

x - 1 (x - 1)( x + 1)

3 = li

(x -m

1)

x 1

3 x - 3lim

x - 1

(x - 1) x 1

x 1

x 1 x 1

3= lim =

( x + 1) ( x + 1)

( x + 3 - 2)( x + 3 + 2) = lim =

(x - 1)( x + 3 + 2)

x + 3 - 4 = lim = lim

(x - 1)( x + 3 + 2

x 1

)

-

x 1

3

2

x + 3 - 2lim

x - 1

(x - 1)

x 1 x 1

=( x + 3 + 2)

1 = =

1+ 3 + 2)

O

3 x - 3 x + 3 - 2 3 1= lim + lim = + =

x - 1 x - 1 2 4 :

x 1

x 1

1

4

3 x + x + 3 - 5 7lim

x - 1 4

x - 1 4lim =

73 x + x + 3 - 5

:

-

.

:

E .

:

-

.

:

1. -

(

-

) .

2.

x - x0 .

3.

.

:

1.

,

-

.

-

( 0 : 0).

2.

-

, -

-

.

009

• 3 2

x 2

x + 5x + 13 - 2x + 5N : lim

x - 2

3 2

x 2 x 2

3 32 2

3 2

lim x + 5x + 13 = 3 lim 2x + 5 = 3

x + 5x + 13 - 2x + 5 x + 5x + 13 - 3 - 2x + 5 + 3= =

x - 2 x

(

- 2

x + 5x + 13 - 3 2x + 5 - 3= -

x - 2 x - 2

=

3)

=

3 2

x 2

x + 5x + 13 - 3lim

x - 2 3 32 2 2 23

32 2 2x 2 3

2

32 2 2x 2 3

x 2

( x + 5x + 13 - 3)( (x + 5x + 13) + 3 x + 5x + 13 + 9)lim =

(x - 2)( (x + 5x + 13) + 3 x + 5x + 13 + 9)

x + 5x + 13 - 27 = lim =

(x - 2)( (x + 5x + 13) + 3 x + 5x + 13 + 9)

(x - 2) = lim

(x + 7)

(x - 2) 32 2 23

2 33

x 2 x 2

x 2 x 2

=( (x + 5x + 13) + 3 x + 5x + 13 + 9)

2 + 7 9 9 = = = =

9 + 9 + 9 27(27) + 3 27 + 9

=

( 2x + 5 - 3)( 2x + 5 + 3) 2x + 5 - 9 = lim = lim =

(x - 2)( 2x + 5 + 3) (x - 2)( 2x + 5 + 3)

2 (x - 2)2x - 4 = lim = lim

(x - 2)( 2x + 5 + 3)

x 2

1

3

2x + 5 - 3lim

x - 2

(x - 2)

3 2

x 2 x 2

2= =

( 2x + 5 + 3) 9 + 3

2 2 = = =

3 + 3 6 :

x + 5x + 13 - 3 2x + 5 - 3= lim - lim =

x - 2 x - 21 1

= - =3 3

3 2

x 2

1

3

x + 5x + 13 - 2x + 5lim

x - 2

0

:

( )

.

:

E .

:

-

.

:

1.

-

,

-

( 0 : 0).

2.

-

.

:

,

-

,

-

.

010

• 6

3 6x 0

N :

x + 1 - x + 1lim

x + 1 - x + 1

6

3

2 3

6 6

x 0

3(1)

2y 1

.. : 6

y = x + 1

O

y = x + 1

y = x + 1 (1)

lim x + 1 = 0 + 1 = 1 y 1

E

y - y= lim =

y - y

6

3 6x 0

x + 1 - x + 1lim

x + 1 - x + 1

y 1

y 1

(y + 1) = lim =

= lim (y + 1) =

y (

y - 1)

y(

=

y

1 =

- 1)

+ 1

2

:

-

.

:

E .

:

-

.

:

1. -

y ,

y .

2. -

y

-

y .

011

• x 1

x 2

x 2

1. f x = 1 :

5f(x) - 2 lim = 2 .

2f(x) - 3

f(x) - 3x + 22. lim :

x - 2f(x) - 4

lim = 7 .x - 2

x 1

1.

5f(x) - 2 h(x) = (1) o lim h(x) = 2 (2)

2f(x) - 3

' (1) :

5f(x) - 2h(x) = 5f(x) - 2 = h(x)(2f(x) - 3)

2f(x) - 3

5f(x) - 2 = 2h(x)f(x) - 3h(x) 5f(x) - 2h(x)f(x) = 2 - 3h(x)

2 - 3h(x)f(x) =

.

(2)

x 1

x 1

5 - 2h(x)

2 - 3h(x) 2 - 3 2 2 - 6 - 4, = lim = = = =

5 - 2h(x) 5 - 2 2 5 - 4 1

2.

f(x) - 4 h(x) = (3) o lim h(x) = 7 (4)

x - 2' (3) :

f(x) - 4h(x) = f(x) - 4 = h(x)(x - 2)

x - 2

x 1lim f(x) - 4

(4)

x 2

x 2

f(x) = h(x)(x - 2) + 4

, = lim [h(x)(x - 2) + 4] = 7 0 + 4 =

f(x) - 3x + 2 0 lim .

x - 2 0

f(x) - 3x + 2 f(x) - 3x + 2 f(x) - 4 - 3(x - 2)= = =

x - 2 x - 2 x - 2

f(x) -

- +

=

4 4

x 2lim f(x) 4

3 (x - 2)4-

x - 2 x - 2

(3)

(4)

x 2

= h(x) - 3

= lim(h(x) - 3) = 7 - 3 =x 2

f(x) - 3x + 2lim 4

x - 2

:

-

f .

:

E -

f .

:

f(x)

.

:

1.

f,

,

h(x)

f(x) .

2.

f(x) .

3. A

-

f,

-

-

,

.

012

• x 4

N :

|x - 4|lim

|x - 4|+ 1 - 1

-

-

x 4

x 4

|x - 4| f(x) =

|x - 4|+ 1 - 1

E

x < 4 x - 4 < 0 |x - 4|= - x + 4

x > 4 x - 4 > 0 |x - 4|= x - 4

-x + 4 = lim =

-x + 4 + 1 - 1

(-x = lim

x 4

|x - 4|lim

|x - 4|+ 1 - 1-

-

-

2x 4

x 4

+ 4)( -x + 5 + 1)=

( -x + 5 - 1)( -x + 5 + 1)

(- x + 4)( - x + 5 + 1) = lim =

( - x + 5) - 1

(- x + 4)( - x + 5 + 1) = lim =

- x + 5 - 1

-

-

+ +

x 4

x 4

x 4 x 4

( - x + 5 + 1) = lim =

= lim - x + 5 + 1 = - 4 + 5 + 1 =

x - 4 (x - 4)

(- x

( x - 3 + 1) = l

+ 4)

-

im = limx - 4 + 1 - 1 ( x - 3 - 1)( x - 3

4

+ 1

x +

x 4

2

|x - 4|lim

|x - 4|+ 1 - 1+

+ +

+ +

2x 4 x 4

x 4 x 4

=)

(x - 4)( x - 3 + 1) (x - 4)( x - 3 + 1) = lim = lim =

x - 3 - 1( x - 3) - 1

( x - 3 + 1)

(x - 4)

x = lim = lim x - 3 + 1 =

4

-

= 4 - 3 + 1 =

,

f x = 4 :

x 4 x 4

x 4

2

lim f(x) = lim f(x) = 2

limf(x) = 2

- +

:

-

f .

:

-

f x0 .

:

( ) .

:

x0 -

:

1. -

f .

2. A

,

-

, -

.

,

-

x0 .

013

• 2

2 3 2 x 0 x 0x

2

o :

(x) x - 3x xlim lim lim

- 2x2x - x x + 2x - x

x 0

x 0

x 0 x 0 x 0

(x) x = lim =

x x(2x - 1)

( ) 1 = lim =

2x - 1

( ) 1 = lim lim lim =

2x - 1

x x

x x

x x

x x

2x 0

(x)lim

2x - x

2

:x

2x 0

2

2

2x 0

2

x 0

1 = 1 1 =

2 - 1

x- 3

x = lim =x + 2x - 1

x- 3

x x = lim =x + 2x - 1

x

= lim

x

x

2

3 2x 0

1

x - 3xlim

x + 2x - x

2

0 = 0

0 = 1

x = ( -x)

2

x

2

- 3x =

x + 2x - 10

1 - 31 = =

0 + 0 - 1- 3

= =- 1

( - x)

2 = lim =

2( - x)2

x

2

3

xlim

- 2x

x

2

1 = lim =

2

1

( -

=

x)2

- x

1

2

=2

1

2

:

-

, -

.

:

-

x0 .

:

-

0 0x x x x

f(x)lim = 1 lim f(x) = 0

f(x)

:

,

,

-

-

-

-

.

014

• 2 x x

2 x + 1 xx 1

N o :

2 3 - 7 3 + 3lim

3 - 7 3 - 6

x

x

x 2 x 3 = u

x 2 xx 1 u 3

0 .

0

3 = u

O x 1 u 3

2 (3 ) - 7 3 + 3= lim =

3 (3 ) - 7 3 - 6

2 x x

2 x + 1 xx 1

2 3 - 7 3 + 3lim

3 - 7 3 - 6

2

2u 3

u 3

u 3

2u - 7u + 3 = lim =

3u - 7u - 6(2u - 1)

= lim =(

(3u + 2)

2u - 1 = lim =

3u + 22 3 - 1

=3

u - 3)

(u - 3)

3

=

+ 2

=5

11

:

f .

:

-

f .

:

N -

.

:

1. -

.

2. ,

y .

3. y,

x x 0

4.

0y y

lim f(y)

.

015

• x

x 0 x e

N o :

e - 1 lnx - 1 lim lim

x x - e

x

x 0

x 0

x

x 0

x 0 x 0

x 0

E

e - 1 x= lim =

x x

x= lim =

x

x= lim =

x

x1

= lim x lim

e - 1lim

=

x

x

1

x

x

x 0

e - 1lim

x

x

x 0

x e

x e x e , u

0 x

1

0

1= 1 =

xlim

x=

E

lnx - lne= lim =

x - ex

lne= lim =

x x lim = 1 lim = 1

x x

x- 1

e

x e

1

lnx - 1lim

x - e

x u =

e

u 1

lnu = lim

u - 1 =

1

:

f .

:

-

f .

:

N -

g(x) ln(g(x))e -1

g(x) g(x) -1

.

:

f

-

.

:

h(x)

x 0

e -1lim = 1, h(x) 0

h(x)

( D.L.H.).

x 1

ln(h(x))lim( ) = 1, h(x) 1

h(x) -1

( D.L.H.).

016

• 2

2

2

2

o ,

x + 2x - x 2

x - 4x = 2 : f(x) = x - x

x > 2x - 3x + 2

-

-

-

+

2

2x 2

x 2

x 2

2

x 2x 2

2

x 2

x 2

E

lim(x - 4) = 0 lim(x + 2x -) = 0 4 + 4 - = 0 (1)

lim f(x)

( lim(x + 2x -) 0 lim f(x) = )

lim (x - 3x + 2) = 0

lim f(x)

+

+

2

x 2

2

x 2x 2

lim x - x) = 0 4 - 2 = 0 (2)

( lim (x - x) 0 lim f(x) = )

, (1) (2) : 4 + 8 - = 0

= 2 = 12

= 2

= 12

(

- - -

2

2

2

2

2

2x 2 x 2 x 2

:

x + 4x - 12 x 2

x - 4f(x) = x - 2x

x > 2x - 3x + 2

(x + 6) (x - 2)x + 4x - 12 lim f(x) = lim = lim

x - 4

(x + 2) (x - 2)

-

+ + +

x 2

2

2x 2 x 2 x 2

=

x + 6 2 + 6 8 = lim = = = 2

x + 2 2 + 2 4

x (x - 2)x - 2x lim f(x) = lim = lim

x - 3x + 2

(x - 1) (x - 2)

+x 2

x 2

=

x 2 2 = lim = = = 2

x - 1 2 - 1 1A lim f(x) = 2

:

-

f

.

:

,

f,

,

.

:

.

:

f

.

1. -

,

.

2.

-

.

3.

-

, -

.

017

• 2

3x 2

2 2

2

N :

x + 2 x 2

limf(x) f(x) = x - 3x + 4 x > 2

x - 1

, x = 3 :

x - x - 10 x < 3 g(x) =

x +

2 x - 1 x > 3

- -

+ +

2

x 2 x 2

3

x 2 x 2

E

lim f(x) = lim (x + 2) = 4 + 2 = 6

x - 3x + 4 8 - 6 + 4lim f(x) = lim = = 6

x - 1 2 - 1

f x = 2

x 2 x 2

x 2

lim f(x) = lim f(x) = 6

limf(

- +

- +

- +

x 3 x 3

2 2 2 2

x 3 x 3

2 2 2 2

2

g x = 3, :

lim g(x) = lim g(x)

lim ( x - x - 10) = lim (x + x - 1)

3 - 3 - 10 = 3 + 3 - 1

9 - 3 - 10 = 9

x) = 6

.

2

2

2

+ 3 - 1

6 - 3 - 18 = 0

2 - - 6 = 0

= 2

3 = -

2

:

-

f .

:

-

f .

:

( ) .

:

1.

f .

2. A

,

, -

.

:

-

,

-

,

-

f

.

018

• 2 2

x 0

x 4

f x

: 4x + x + 1 f(x) x + x.

: limf(x)

A x > 0 : 4 x f(x) x + 4, :

limf(x)

x 4

f(x) - 8 lim

x - 4

2 2

x 0 x 0 x 0

x 0 x 0

= lim 4x + lim x + lim1 = 0 + 0 + 1 =

= lim x + lim x = 1+ 0 =

, :

2 2

x 0

x 0

x 0

lim (4x + x + 1) 1

lim (x + x) 1

limf(x) = 1

= 4 4 = 4 2 =

= 4 + 4 =

, :

4 x f(x) x + 4 4 x - 8 f(x) - 8 x - 4

x < 4 :

4 x - 8 f(x) - 8 x - 4

x - 4 x - 4 x

x 4

x 4

x 4

lim 4 x 8

lim (x + 4) 8

limf(x) = 8

x 4 x 4 x 4

4 x - 8 f(x) - 8= 1 1

- 4 x - 4 x - 4 x > 4 :

4 x - 8 f(x) - 8 x - 4 4 x - 8 f(x) - 8 = 1 1

x - 4 x - 4 x - 4 x - 4 x - 4

4 x - 8 4( x - 2)( x + 2) 4 lim = lim = lim =

x - 4 (x - 4

(x - 4)

)( (xx + 2) - 4)( x + 2)

4 = = 1

2 + 2

. :

A, :

x 4 x 4

x 4

f(x) - 8 f(x) - 8lim = lim = 1

x - 4 x - 4

f(x) - 8lim = 1

x - 4

- +

:

f .

:

-

f

f

x 0 .

:

-

.

:

1. -

-

f

-

f -

.

2. -

.

3. -

, -

,

.

:

,

,

-

, :

(

1).

1 .

019

• x 2

x 2 x 2

x g(x) - 2 :

g(x) - 24 g(x) + 2 f(x) g(x) + 6 lim = 1, :

x - 2f(x) - 8

limf(x) limx - 2

x 2

x 2 x 2 x 2 x 2 x 2

g(x) - 2 h(x) = o lim h(x) = 1

x - 2g(x) - 2

h(x) = (x - 2) h(x) = g(x) - 2 g(x) = (x - 2) h(x) + 2 x - 2

limg(x) = lim[(x - 2) h(x) + 2] = lim(x - 2) lim h(x) + lim 2 =

x 2 x 2

x 2 x 2 x 2

4 g(x) + 2

= 0 1+ 2 = 2

lim[4 g(x) + 2] = 4 limg(x) + 2 = 4 2 + 2 = 8 E,

lim[g(x) + 6] = limg(x) + lim 6 = 2 + 6 = 8

, :

f(x) g(x

x 2 limf(x) = 8

x 2 x 2

4( g(x) + 2

4 g(x) + 2

2 g(x) + 2

4( g(x) + 2

4( g(x) + 2 g(x) + 2

) + 6 - 2) f(x) - 8 g(x) + 6 - 8

- 8 f(x) - 8 g(x) - 2

- 2) g(x) - 2f(x) - 8x > 2 :

x - 2 x - 2 x - 2

- 2)g(x) - 2 f(x) - 8x < 2 :

x - 2 x - 2 x - 2

- 2 4( lim = lim

x - 2

x 2

x 2

g(x) + 2

g(x) + 2

g(x) - 2

g(

g

x) +

(x) + 2

g(x) +

2

2

- 2)=

(x - 2)

4( - 4) = lim =

(x - 2)( + 2)

4( ) = lim =

(x - 2)( + 2)

( + 2)

( + 2

)

x 2 x 2

x 2

g(x) - 2

g(x) + 2

g(x) - 2

2 + 2 4

4 = lim lim =

x - 2 + 2

4 4 4 = 1 = = = 1 = lim

4 x - 2+ 2 + 2

. :

x 2 x 2

f(x) - 8 f(x)lim = lim

x - 2

- +

A, : x 2

- 8= 1

x - 2

f(x) - 8lim = 1

x - 2

:

f

g, -

-

g.

:

-

f

f x 0 .

:

-

.

:

1. h(x) -

g -

(

h(x)) .

2.

g(x) .

3. -

g(x) .

4. g(x)

-

.

5. -

, -

,

.

:

-

f,

-

-

...

.

020

• x 0

f(x)A |f(x) - x| 1 - 2x : lim = 1 .

x

2 2

2

2 2 2 2

: 2x = 1- 2 x 2 x = 1- 2x

|f(x) - x| 1- 2x |f(x) - x| 2 x

-2 x f(x) - x 2 x x - 2 x f(x) x + 2 x (1)

(1)

+ +

2 2

2 2

2

x 0 x 0

x.

x x x xf(x) A x > 0 (1) : - 2 + 2

x x x x x

x x x xf(x)- 2x + 2x

x x x x x

x x x lim - 2x lim - 2 lim

x x x

=

+ +

+ + + +

2

x 0 x 0

2

2 2

x 0 x 0 x 0 x 0

xx lim =

x

= 1- 2 0 1 = 1

x x x x lim + 2x = lim + 2 lim x lim =

x x x x

2

2 2

= 1+ 2 0 1 1

, :

x x x xf(x) A x < 0 (1) : + 2 - 2

x x x x x

+x 0

f(x)lim = 1

x

=

- - - -

2 2

2 2

x 0 x 0 x 0 x 0

x x x xf(x) + 2x - 2x

x x x x x

x x x x lim + 2x = lim + 2 lim x lim =

x x x x

- - - -

2

2 2

x 0 x 0 x 0 x 0

2

= 1+ 2 0 1 1

x x x x lim - 2x lim - 2 lim x lim =

x x x x

= 1- 2 0 1 1

=

=

=

, :

E

-x 0

x 0

f(x)lim = 1

x

f(x)lim = 1

x

:

f.

:

-

f

f x 0 .

:

-

.

:

1. -

-

f

-

f -

.

2. -

-

.

3. -

, -

,

.

:

, -

,

f(x)

x -

x, -

x < x 0 x > x 0

( ) .

x

x -

x 0

xlim = 1

x

021

• 2 2

2 2x x

x x x x

g(x) - 2f(x) - 3 lim [( ) + ( ) ] = 0, :

x + 3 x + 2lim f(x) lim g(x)

0 0

0

2 2 2

2 2 2

2 2 2

2 2 2x x x x

x x

g(x) - 2f(x) - 3 f(x) - 3 0 +

x + 3 x + 3 x + 2

g(x) - 2f(x) - 3 f(x) - 3 0 lim lim +

x + 3 x + 3 x + 2

f(x) - 3 0 lim

x

0 0

0

2

2

2

2 2x x x x

2

2

x x

0+ 3

, :

f(x) - 3 f(x) - 3lim = 0 lim = 0

x + 3 x + 3

f(x) - 3 = h(x)

x + 3

f(x) = (x + 3)h(x) + 3 lim h(x)

0 0

2

0

2 2 2

2 2 2

2 2

2 2 2x x x x

= 0

, = (x + 3) 0 + 3 =

g(x) - 2 g(x) - 2f(x) - 30 +

x + 2 x + 3 x + 2

g(x) - 2 g(x) - 2f(x) - 3 0 lim lim +

x + 2 x + 3 x + 2

0x x

lim f(x) 3

0

0 0

2

2

2x x

2

2 2x x x x

2

g(x) - 2 0 lim 0

x + 2

, :

g(x) - 2 g(x) - 2lim = 0 lim = 0

x + 2 x + 2

g(x) - 2 = r(x)

x + 2

0

2

x x

2

0

g(x) = (x + 2)r(x) + 2 lim r(x) = 0

, = (x + 2) 0 + 2 =

0x x

lim g(x) 2

:

f (

f, g) .

:

-

f ( g)

x0 .

:

N -

-

.

:

1. -

0 f(x) f(x) + g(x),

f(x) > 0 g(x) > 0

0 f 2(x) f 2(x) + g 2(x),

.

(f(x) + g(x))

(f 2(x) + g 2(x))

.

-

f(x)

f 2(x) .

2. f(x) -

f(x), -

:

h(x)

i

f(x)

o o f(x)

(

h(x) 0) .

:

-

g ( ) -

.

022

• =

3 2

2x 0 x 0

f : f (x) + f(x) = x ,

x ,

f(x) lim f(x) , lim

x

2

2 3 2 2 2

2

2

2

2f (x)+ 1 0

, x , :

x f (x) + f(x) = x f(x)[f (x) + 1] = x f(x) = (1)

f (x) + 1

f (x) + 1 1

' (1) ,

x |f(x)|= =

f (x) + 1

2 2 2 f (x)+ 1 1x 0 2 2

2

2 2

2

x 0

2

x 0

(1)

2x 0

x x |f(x)| x

f (x) + 1

- x f(x)| x

lim (- x ) = 0

lim x = 0

E

f(x) = lim = li

x

x 0lim f( x) = 0

2

2 2x 0 x 0

x 0

2

2

1 1f (x) + 1m = lim = =

f (x) + 1 [lim f(x)] + 1

1 = =

0 +

x

x

1

1

:

-

f .

:

f

x 0 .

:

N -

-

.

:

1.

f(x) .

2. -

-

(1.) .

3. -

-

|f(x)| g(x), g(x) > 0

4. - g(x) f(x) g(x)

- g(x), g(x) .

.

023

• x 1

x 3

x f(x - 2) = f(x) lim [f(x) - 3x - 2] = 5

: limf(x) .

x 1

x 1

x 1 x 1 x 1 x 1

lim [f(x) - 3x - 2] = 5 f(x - 2) = f(x) (1)

h(x) = f(x) - 3x - 2 lim h(x) = 5 (2)

f(x) = h(x) + 3x + 2

lim f(x) = lim [h(x) + 3x + 2] = lim h(x) + 3lim x + li

(2)

x 1

y 1

y = x - 2(1) (3)

x 3 x 3 y 1 y 1

m 2 =

= 5 + 3 + 2 = 10

A x = y : lim f(y) = 10 (3)

= lim f(x - 2) = lim f(y) =

x 3 limf(x) 10

:

f x 1 .

:

f

x 2 .

:

,

( x 1 ) -

f (

x 2 ) .

:

1. h(x)

f(x) .

2. f(x) .

3. -

f(x) x 1 .

4. y = x (x 2 x 1),

x 1 < x 2 ( x x 2

y x 1)

5. -

,

.

024

• x 2

2

2x - 2 x 2 x 2

f : lim f(x) = 1.

,

f (x) - f(-x) lim f(x) lim f(x - 4) lim

f (x) + 3 - 2f(x - 4)

x 2

x - 2 x - 2

x - 2 x - 2

lim f(x) = 1 (1)

f(x) = f(- x) lim f(x) = lim f(- x), x , f

.

u = - x lim u = lim(- x) = 2, u

(1)

x 2

x 2 x 2

( )

x - 2

2

, = lim f(u) =

u = x - 4 lim u = lim(x - 4) = - 2, u - 2

, = lim f(u) =

x - 2

x 2

lim f(x) 1

limf(x - 4) 1

2

2x 2

x 2 x 2 x 2

f (x) - f(x) lim

f (x) + 3 - 2f(x - 4)

x 2 : x - u = - x f(x - u) = f(- x)

u = f(x), lim u = lim(x) = 1 = lim(x - 4), u 1

o

2 2 2

2 2u 1 u 1

2 2

2 2 2u 1

u

2

,

u - u (u - u) lim = lim =

u + 3 - 2u ( u + 3 - 2u)

(u - u)( u + 3 + 2u) = lim =

( u + 3) - (2u)

( u +

3 + 2

u)

( u + 3 + 2

= lim

u)

2 2

21

2

u 1

2

u 1

(u - u)( u + 3 + 2u) =

- 3u + 3

u ( u + 3 + 2u) = lim =

- 3 (u + 1)

u( u + 3 + 2u)

(u - 1)

(u - 1)

= lim = - 3(u + 1)

2-

3

:

f x 1

f / .

:

f

x 2 .

:

,

( x 1 ) -

f (

x 2 ) .

:

1. u = - x

f(- x) = f(x) / f(- x) = - f(x)

( / )

0 0x x x x

lim f(x) = lim f(- x)

/

0 0x x x x

lim f(x) = - lim f(- x)

( / )

f(x) .

2. -

f(x) x 1

-

.

025

• 2

2x 0

( x) : lim .

x

2

2

22 2 2

2 2 2

2 u = x

2x 0 x 0 u 0 x 0 ( 0 = 0)

x 0

2

2

( x) ( x) ( x) x = =

x x x x

( x) (u) lim = lim = 1 (1)

u x

x lim

x

x

2 2

2

x 0

22 (1)

2x 0 x 0 (2)

x = = 1 = 1 (2)lim

x x

( x) x= lim lim = 1 1 =

x x

2

2x 0

( x) lim 1

x

:

-

f g .

:

-

f g .

:

-

g -

, -

f

.

:

1. u = g(x) .

2. B -

g(x) x 0 ,

u 0 .

3. -

f(u) u 0 .

4. A g(x) u 0 x 0 ,

:

x x u ulim f(g(x)) = lim f(u) 0 0

026

• ,

0

0 4

( ) f x o :

2x - 3f(x) = x = 1 .

4 (x - 1)

4

4

4

x 1

(x - 1) > 0

4x 1

x 1 ((x - 1) > 0 1) :

2x - 3 1f(x) =

4 (x - 1)

2x - 3 2 1 - 3 1 lim = = - (1)

4 4 4

1 lim = + (2)

(x - 1)

lim(1)

4x 1 x 1 (2)

2x - 3 1 1= lim lim = - (+ ) =

4 4(x - 1)

4x 1

2x - 3 - 4 (x - 1)

:

0 0x x x x

f(x)lim lim g(x) = 0

g(x)

(x - x 0) -

x 0 .

:

.

:

-

g

-

.

:

1. g, -

g(x) = (x - x 0) h(x)

0x x

f(x)lim

g(x)

0 0x x x x

0

1 f(x)lim lim

x - x h(x) .

2. B 0x x

f(x)lim

h(x)

3. 0x x

0

1lim

x - x.

x - x 0 > 0 : 0x x

0

1lim = +

x - x

x - x 0 < 0 : 0x x

0

1lim = -

x - x

:

-

-

x xlim f(x) = 0

x xlim g(x) =

0

024

028

• ,

0

2

0

( ) f x o :

x + 3x - 2f(x) = x = 0 .

x |x|

+

2

2 2

x 0

x

x 0 (x |x| -

0) :

1f(x) = (x + 3x - 2)

x |x| x > 0

lim (x + 3x - 2) = 0 + 3 0 - 2 = - 2

lim

+ +

+ +

-

x > 0

20 x 0

2

x 0 x 0

2

x 0

1 1 = lim = + x |x| x

1 = lim (x + 3x - 2) lim = - 2 (+ ) =

x |x| x < 0

lim (x + 3x

+

2

x 0

x + 3x - 2lim -

x |x|

- -

- - -

2

x < 0

2x 0 x 0

2

x 0 x 0

- 2) = 0 + 3 0 - 2 = - 2

1 1 lim = lim = - -

x |x| x

1 = lim (x + 3x - 2) lim =

x |x|

2

x 0

x + 3x - 2lim

x |x|

- +x 0 x 0

0

- 2 (- ) =

, lim f(x) lim f(x)

f

x = 0 .

+

:

0 0x x x x

f(x)lim lim g(x) = 0

g(x)

(x - x 0) -

x 0 .

:

.

:

-

g

-

.

:

1. g, -

g(x) = (x - x 0) h(x)

0x x

f(x)lim

g(x)

0 0x x x x

0

1 f(x)lim lim

x - x h(x) .

2. B 0x x

f(x)lim

h(x)

3. 0x x

0

1lim

x - x.

:

x - x 0 > 0 : 0x x

0

1lim = +

x - x

x - x 0 < 0 : 0x x

0

1lim = -

x - x

:

-

-

x xlim f(x) = 0

x xlim g(x) =

0

029

• ,

0

0 3 2

( ) f x o :

2x - f(x) = x = 1 .

x - 2x + x

2

3 2 2

(x - 1) > 0

2x 1 x 1 x 1

2x - 2x - f(x) = = =

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

2x - 1 lim f(x) = lim lim = (2 - ) (+ )

x (x - 1)

:

2 - > 0

2x 1 x 1

= (2 - ) (+ ) =

2 - < 0

= (2 - ) (+ ) =

2 - = 0

2x - 2lim f(x) = lim

x (x - 1)

x 1

x 1

lim f(x) +

lim f(x) -

< 2

> 2

< 2

> 2

= 2

0 2

0 0

2x 1 x 1

x 1 x 1 x 1

2 (x - 1) 2= lim = lim =

x (x - 1)x (x - 1)

2 1 1 = lim lim = 2 lim

x x - 1 x - 1 ( x - 1 1)

- -

+ +

- +

x 1 x 1

x 1 x 1

x 1 x 1

1 x - 1 < 0 : lim f(x) = 2 lim = 2 (- ) = -

x - 11

x - 1 > 0 : lim f(x) = 2 lim = 2 (+ ) = + x - 1

, lim f(x) lim f(x)

0

f

x = 1 .

:

0 0x x x x

f(x)lim lim g(x) = 0

g(x)

f .

:

.

:

-

g

-

.

:

1. g, -

g(x) = (x - x 0) h(x)

0x x

f(x)lim

g(x)

0 0x x x x

0

1 f(x)lim lim

x - x h(x) .

2. B 0x x

f(x)lim

h(x),

-

.

3. 0x x

0

1lim

x - x.

x - x 0 > 0 : 0x x

0

1lim = +

x - x

x - x 0 < 0 : 0x x

0

1lim = -

x - x

:

-

-

x xlim f(x) = 0

x xlim g(x) =

0

030

• x 1

x 1

2

x 1

lim f(x), o :

x - 4lim = +

f(x)

lim [f(x) (3x - 2) = +

x 1 x 1

x 1

x - 4 h(x) = 1 .

f(x)

x - 4 1 h(x) f(x) = x - 4 f(x) = = (x - 4)

h(x) h(x)

1 lim h(x) = + lim = 0

h(x)

lim(

x 1 x 1

2

2 2

x - 4) = - 3

1 = lim(x - 4) lim = - 3 0 =

h(x)

g(x) = f(x)(3x - 2) 1 .

g(x) 1 f(x) = = g(x)

3x - 2 3x

x 1lim f(x) 0

x 1

2 2x 1

2x 1 x 1

- 2 lim g(x) = +

1 1 lim = = 1 0

3x - 2 3 1 - 2

1 = lim g(x) lim = (+ ) 1 =

3x - 2

x 1lim f(x) +

:

f,

.

:

f .

:

h(x)

f(x) = g(x)

, o

g, h -

, -

.

:

1. h(x) -

.

2. f(x) .

3. , -

.

:

-

-

x xlim f(x) = 0

x xlim g(x) =

0

031

• 0

N

f

x = - , +

-

.

0

x -

0

x +

x = -

lim f(x) = +

x = +

lim f(x) = 1

:

-

f .

:

-

f, + - .

:

-

C f

+

- .

:

x + ,

C f ( -

yy) .

x - ,

C f

( yy) .

033

y

3

2

1

0 x

• 5 2

x +

2 2 + 1

x -

5 2

x +

:

lim (2x + 3x + x + 1)

lim [(x - 1) + (x + 1) ]

lim (( - 1)x + ( + 1)x + x + 1))

5

x +

x -

2 v + 1

2

x +

= lim (2x ) =

lim (x ) =

= 1 :

= lim (2x ) =

5 2

x +

2 2 + 1

x -

5 2

x +

lim (2x + 3x + x + 1) +

lim [(x - 1) + (x + 1) ] -

lim (( - 1)x + ( + 1)x + x + 1)) +

=

5

x +

:

1 :

= ( - 1) lim x ) =

T

=

5 2

x +

5 2

x +

lim (( - 1)x + ( + 1)x + x + 1)) +

lim (( - 1)x + ( + 1)x + x + 1)) +

(

:

.

:

-

.

:

-

-

.

:

1.

.

2. -

-

-

, -

:

-

.

-

.

034

• 5 2

2x -

5 2

5x +

3

4 2x -

:

2x + 3x + x + 1lim

3x + x + 1

2x + 3x + x + 1lim

x + x + 1

x - x + 1lim

2x + x + 1

5

2x -

x -

x

3

-

3

2x = lim ( ) =

3x2

= lim ( x ) =3

2 = lim (x ) =

3

5 2

2x -

2x + 3x + x + 1lim

3x + x + 1

5

5x +

3

4x -

x -

2 = (- ) =

3

2x = lim ( ) =

x

x lim ( ) =

2x1 1

= lim ( ) =2 x

5 2

5x +

3

4 2x -

2x + 3x + x + 1lim 2

x + x + 1

x - x + 1lim

2x + x + 1

-

=

1 = 0 =

2 0

:

-

.

:

.

:

-

.

:

1. -

-

.

2. :

-

-

-

,

+ - .

-

-

-

-

,

0 .

-

-

, -

.

035

• 3 2

2x -

, :

( + - 5)x + ( - 1)x + 2lim = 2

( - 1)x + x + 1

3 2 3

2 2x x

3 2

2x

1

( + - 5)x + ( - 1)x + 2 ( + - 5)x + - 5lim = lim = (- )

- 1( - 1)x + x + 1 ( - 1)x

+ - 5 0,

- 1

( + - 5)x + ( - 1)x + 2 lim = ,

( - 1)x + x + 1

(

2

2x x 2

2

2).

+ - 5 = 0 + - 5 = 0 (1)

- 1

( (1)) :

( - 1)x + 2 ( - 1) - 1lim = 2 lim = 2 = 2

- 1( - 1)x + x + 1 ( - 1)

- 1 = 2 - 2 (2)

,

x

x

+ = 5

- 2 = - 1

(+)

(1) (2)

+ = 5 2 + 2 = 10 3 = 9 = 3

- 2 = -1 - 2 = -1 - 2 = -1 3 - 2 = -1

= 3

= 2

:

-

.

:

.

:

-

.

:

1. -

-

.

2. :

-

-

-

,

+ - .

-

-

-

-

,

0 .

-

-

, -

.

3. -

()

-

-

.

036

• 2 2

x -

:

lim ( 9x - x + 1 - 4x + 2x + 1)

2 2

2 2x -

x < 0

2 2x -

x -

x -

,

=

1 1 2 1 = lim x (9 - + ) - x (4 + + ) =

x xx x

1 1 2 1 = lim |x| 9 - + -|x| 4 + +

x xx x

1 lim - x 9 -

2 2

x -

x < 0 .

lim ( 9x - x + 1 - 4x + 2x + 1)

2 2

2 2x -

2 2x - x -

1 2 1+ + x 4 + + =

x xx x

1 1 2 1 = lim - 9 - + + 4 + + =

x xx x

1 1 2 1 = lim x lim - 9 - + + 4 + + =

x xx x

= - (- 9 - 0 + 0 + 4 + 0 + 0) =

= - (- 9 + 4) =

= - (-1) =

=

+

:

-

.

:

-

( ) .

:

-

.

:

1.

x -

( ).

2. -

x

1lim f(x) = 0

x .

3. -

-

, -

... -

.

:

x,

x + x > 0 |x| = x

x - x < 0 |x| = - x

037

• 2 2

x +

:

lim ( 16x + 8x + 4x - 1 - 6x)

2 2

x +

2 2

x + x +

2

x +

4x

x +

,

= lim ( 16x + 8x - ) + ( 4x - 1 - ) =

= lim ( 16x + 8x - 4x) + lim ( 4x - 1 - 2x) =

( 16x + 8x - 4x)( 1 = l

2x

im

2 2

x +

x > 0 .

lim ( 16x + 8x + 4x x- 1 - =6 )

2

2

2 2

2x +

2 2 2 2

2x + x + 2

2x + x +

6x + 8x + 4x)+

16x + 8x + 4x

( 4x - 1 - 2x)( 4x - 1 + 2x) + lim =

4x - 1 + 2x

16x + 8x - 16x 4x - 1- 4x = lim + lim =

8 ( 4x - 1 + 2x)x (16 + ) + 4x)

x

8x - 1 = lim + lim

8 ( 4x - 1 + 2|x| 16 + + 4x

x

x > 0

2x + x +

2x + x +

x)

8 - 1 = lim + lim =

8 ( 4x - 1 + 2x)( 16 + + 4)

x

8 - 1 = lim + lim =

8 ( 4x - 1 + 2x)16 + + 4

x

8 = + 0 =

16 + 0 + 4

8

x

= =8

x

1

:

-

.

:

-

( ) .

:

-

.

:

1.

x -

( ).

2. -

x

1lim f(x) = 0

x .

3. -

-

, -

... -

.

:

x,

x + x > 0 |x| = x

x - x < 0 |x| = - x

( -

-

),

, ,

(

) .

038

• 2

2

x + x

4x + 2x + 3 + 3x + 2 : f(x) =

x + x + 1 + 4x + 3

: lim f(x) lim f(x)

2

2x +

2

2

x + 2

2

2

x +

x +

4x + 2x + 3 + 3x + 2 = lim =

x + x + 1 + 4x + 3

2 3x (4 + + ) + 3x + 2

x x = lim =1 1

x (1+ + ) + 4x + 3x x

2 3|x| 4 + +

x x = lim

x +

x > 0

lim f(x)

x > 0

2

2

x +

2

2

x +

2

+ 3x + 2 =

1 1|x| 1+ + + 4x + 3

x x

2 3 2x 4 + + + 3 +

x xx = lim =

1 1 3x 1+ + + 4 +

x xx

2 3 24 + + + 3 +

x xx = lim =1 1 3

1+ + + 4 +x xx

4 + 0 + 0 + 3 + 0 =

1+ 0 + 0 +

2x < 0

x -

2

4 + 3= =

4 + 0 1 + 4

x - ...

2 3 2- x 4 + + - 3 -

x xx 4 - 3 - 1= ... = lim = ... = = =

- 31 - 41 1 3- x 1+ + - 4 -

x xx

x -

1

x < 0

1lim

3

:

-

.

:

-

() .

:

-

.

:

1.

x -

( )

-

.

2. -

-

.

3. -

x

1lim f(x) = 0

x .

:

x,

x + x > 0 |x| = x

x - x < 0 |x| = - x

039

• 2 2

x +

f(x) = x + 2x + 3 + 4x + 4x + 5 + x +.

, lim f(x) = 6.

2

2 2x +

x + x +

(x + ), ( x ) :

2 3 4 5= lim x 1+ + + 4 + + + + =

x x xx x

3 + 0 lim f(x) = , ( lim f(x) = 6)

3 + = 0

x +

x > 0

lim f(x) + (3 + ).

2 2

2 2

x > 0

2 2 x x

= - 3

= x + 2x + 3 + 4x + 4x + 5 - + =

= ( x + 2x + 3 - ) + ( 4x + 4x + 5 - ) + =

2x + 3 4x + 5 = + + =

x + 2x + 3 + x 4x + 4x + 5 + 2x

=

3x

x 2x

2

= - 3

f(x)

32 +

x +2 3

1 + + + 1x x

x

A

2 4lim f(x) = 6 + + = 6

1+ 1 2 + 2

2

54 +

x +4 5

4 + + + 2x x

= 4

:

-

( -

) .

:

() .

:

-

.

:

1.

x -

( ). 2. -

x

1lim f(x) = 0

x .

3. -

-

, -

... -

4. -

()

()

-

.

:

x,

x + x > 0 |x| = x

x - x < 0 |x| = - x

040

• x + x +

2

2 3x +

2x +

f(x) lim = 3 lim (3f(x) - x) = 2 :

x

xf(x) + 5x - 2x + 11lim = 4

3x f(x) - x + 3x + 1

2f(x) - 2x - 1lim = 2

3xf(x) - x + 3x

2( x ) 2

x +

2

E

f(x) 2 11+ 5 - +

x x x = lim =3 1

(3f(x) - x) + +x x

3 + 5 - 0 + 0 = =

2 + 0 + 0

2

2 3x +

xf(x) + 5x - 2x + 11lim

3x f(x) - x + 3x + 1

( x)

x +

=

f(x) 12 - 2 -

x x = lim =3

(3f(x) - x) +x

6 - 2 - 0 = =

2 + 0

2x +

4

2f(x) - 2x - 1lim

3xf(x) - x + 3x

= 2

:

-

f .

:

E -

f .

:

-

f -

.

:

1. -

,

, -

-

,

.

2.

f(x) .

041

• 2

2

x + 2

f : , :

3f(x) + f(- x) = x + x, x .

3f(x) + x + x

4 : lim .7

2f(x) - 1 + x2

2

2 2

2 2

x = - x, :

3f(- x) + f(x) = x - x.

f(- x) + 3f(x) = x + x (- 3 ) - 3f(- x) - 9f(x) = - 3x - 3x

3f(- x) + f(x) = x - x 3f(- x) + f(x) = x - x

- 8f(x) = - 2

2 2

2 2

x + 2 2

2

2x +

1 1x - 4x f(x) = x - x

4 2

1 1 3x - x + x + x

4 2 4= lim =1 7

x - x - 1+ x2 2

1x + x

2 = lim =4x - x - 1

=

2

x + 2

3f(x) + x + x

4lim7

2f(x) - 1 + x2

2

x + 2

2

x +

2

1x 1+

2xlim =

1 1x 4 - -

x x

11+

2x = lim =1 1

4 - -x x

1+ 0 = =

4 - 0 - 0

=

1

4

:

f(x), f(-x) .

:

E

f .

:

-

f

.

:

1. -

, (1), -

x - x,

,

(2) .

2. f(-x)

(1) (2),

-

f .

3.

, -

f(x) .

042

• x + x +

x + x +

f,g (,+ ), > 0

lim ((x - 2)f(x) - (2x + 1)g(x)) = 5, lim ((x + 1)f(x) - (2x + 3)g(x)) = 4

, :

lim f(x) lim

g(x)

2

:

(x - 2) - (2x + 1) = = (x + 1)(2x + 1) - (x - 2)(2x + 3) =

(x + 1) - (2x + 3)

D

= 2x

(x - 2)f(x) - (2x + 1)g(x) = h(x)

(x + 1)f(x) - (2x + 3)g(x) = p(x)

+ x + 2x 2+ 1- 2x - 3x

f(x)

g(x)

( x)f(x)

4x + 7

D h(x)(-2x - 3) - p(x)(-2x - 1)

D p(x)(x

+ 4x + 6 =

h(x) -(2x + 1) = =

p(x) -(2x + 3)

(x - 2) h(x) = =

(x + 1) p(x)

E

D h(x)(-2x - 3) - p(x)(-2x - 1)= =

D 4x

- 2) - h

+

(x)(x +

7

1)

3 1h(x)(-2 - ) - p(x)(-2 - )

x xf(x)7

4 +x

g(x)

=

( x)g(x)

D p(x)(x - 2) - h(x)(x + 1)= = =

D 4x + 7

O

3 1h(x)(- 2 - ) - p(x)(- 2 - ) 5(- 2) - 4(- 2)x x = = =

7 44 +

x2 1

p(x)(1- ) - h(x)(1+ ) 4 - 5x x = = =7 4

4 +x

x +

x +

2 1p(x)(1 - ) - h(x)(1 + )

x x7

4 +x

1lim f(x) -

2

1lim g(x) -

4

:

-

f .

:

E -

f .

:

f

.

:

1.

f,

,

h(x)

f(x) .

2.

f(x) .

3. A

-

f,

-

, -

.

:

f, g (

f, g), -

f, g, -

, -

h(x), p(x) -

-

,

f(x), g(x) .

.

043

• 2 3 2

4 4x + x +

N :

6x + x - 22x x x + x x + 2lim lim

3x + x x + x + x

x

( x)

x +

x + lim

E

x 2xx 6 + x - 4

x 2x= lim =

xx 3 +

x

x 2x6 + x - 4

x 2x = limx

3 +x

2

x +

6x + x - 22xlim

3x + x

=x +

+

4

xlim = 0

x

x = 0

x

4

2 4( x )

x + 4

6 + 0 + 0 = =

3 + 0

E

xx 2x + +

x x x= lim

x 1+

3 2

4 4x +

2

x x + x x + 2lim

x + x + x

x +

x +

4

4 3

xlim = 0

x 4

4 xx + lim = 0

x 3

= x 1

+x x

xx 1 2+ +

x x x x = lim x 1

1+ +x x

=

0 + 0 + 0 = =

1+ 0 + 00

:

-

, -

.

:

-

x0 .

:

-

x + x +

x xlim = 0 lim = 0

x x

:

,

,

-

-

-

-

.

044

• x + 2 x + 1 x + 2 x + 1

x x + 1 x x + 1x + x

N :

3 2 - 8 3 + 2 3 2 - 8 3lim lim

4 3 + 3 2 - 1 4 3 + 3 2

x + 1,

0 ( - -

).

x + 2 x + 1

x +

3 2 - 8 3 + 2lim

4

xx x ( 3 )

x xx +

x x

x xx +

3 4 2 - 8 3 3 + 2= lim =

4 3 + 3 2 2 - 1

2 112 - 24 + 2

3 3 = lim =

2 14 + 6 -

3 3

x x + 13 + 3 2 - 1

12 0 - 24 + 2 0 = =

4 + 6 0 - 1 0

x - 1,

0 ( - -

).

- 6

x x x ( 2 )

x xx -

x

xx -

3 4 2 - 8 3 3= lim =

4 3 + 3 2 2

312 - 24

2= lim =

34 + 6

2

x + 2 x + 1

x x + 1x -

3 2 - 8 3lim

4 3 + 3 2

12 - 24 0 = =

4 0 + 6

2

:

f .

:

-

f .

:

N -

.

:

x + :

-

1,

0 .

( -

) .

xx +

:

0 < < 1 lim = 0

x - :

-

1,

0 .

( -

) .

xx -

:

> 1 lim = 0

045

• 1 1

- x x x x

x N o : lim [e + e ]

x x

1 1 - x x

x x

x x

1 1lim - x lim x x x

- lim

= lim e + lim e =

= e + e =

= e

1 1 - x x

x x

x lim [e + e ]

x

x

x

x

1

xlim x

x

1

xlim 1

xx

lim x

1

0

1

x+ e =

1 = + e =

e1

= + e =e

=

1 + e

=

:

-

g(x)f(x) = .

:

-

f .

:

N

g(x) .

:

g(x)

x lim f(x) f(x)

x lim =

x x

x + x +

x x

x + x +

x x

0 < < 1

lim = 0 lim = +

> 1

lim = + lim = 0

x x lim = 0 lim = 0

x x

046

x

1 x 0

x1

x, lim = 1

1

x

• x + 2 x + 2

x + 1 x + 1x

- 2N : lim , 2 .

- 2

x + 2 x + 2 2 x 2 x 2 x x

x + 1 x + 1 x x x x

x + 2 x + 2 2 x x

x + 1 x + 1 x xx - x -

- 2 - 2 2 - 4 2 : = =

- 2 - 2 2 - 2 2

< 2

- 2 - 4 2lim = lim =

- 2 - 2 2

x x 2

x x

x xx -

x x

x

2

xx -

x

x -

2 - 4

= lim = 2

- 2

2 - 4

2 l

im = 0,

=

x -

lim =2

- 2

2

x -

x + 2 x + 2 2 x x

x + 1 x + 1 x xx - x -

- 4 0 = lim =

- 2 0

> 2

- 2 - 4 2lim = lim =

- 2 - 2 2

2 > 1

x

x

x -

x 2

x x

x xx -

x x

x

2

xx -

2 - 4

2 2 = lim = 2

- 22 2

- 4

2 = li

lim = 0, x -

m =

- 22

>

12

2

2

x -

0 - 4 = lim = 2

0 - 2

:

f

() .

:

-

f .

:

N

(-

) x .

:

-

( x)

x .

x + :

-

1,

0 .

( -

) .

xx +

:

0 < < 1 lim = 0

x - :

-

1,

0 .

( -

) .

xx -

:

> 1 lim = 0

047

• x + 1

x N : lim [ln(e - 1) - x]

x + 1

x

x + 1 x

x

x + 1

xx

E

= lim [ln(e - 1) - x] =

= lim[ln (e - 1) - lne ] =

e - 1 = lim[ln ] =

e

x + 1

x lim [ln(e - 1) - x]

x + 1

x xx

xx

x

x

x

e 1 = lim[ln ( - )] =

e e1

= lim[ln (e - )] = e1

= ln[lim (e - )] =e

1e - > 0

e

= ln (e - 0) =

= lne =

= 1

:

-

f .

:

-

f .

:

N -

ln(g(x)) .

:

g(x) .

A g(x)

> 0

x x lim ln[g(x)] = ln[ lim g(x)] =

= ln

0 + x lim ln[g(x)] = -

.

x lim ln[g(x)] =

.

:

x = lne x x

x = e lnx x > 0

x 1lim(lnx) = 0

.

+ x 0

lim (lnx) = -

048

• 1

2x + 1 x - 1

x x 1

N o :

x + 4lim ( ) lim x

x + 3

2x + 1

x

2(x + 3) - 5

x

2(x + 3)

x

E

x + 3 + 1= lim =

x + 3

1= lim 1+ =

x + 3

1 1= lim 1+ 1+

x + 3 x + 3

2x + 1

x

x + 4lim

x + 3

- 5

2 x + 3 - 5

x x

2 x + 3 - 5

x x

lim

=

1 1= lim 1+ lim 1+ =

x + 3 x + 3

1 1= lim 1+ lim 1+ =

x + 3 x + 3

x

2 - 5

1 x 1 x - 1

x 1 x

1=0

x

-

+3

1 0

= e (1+ 0) =

=

E

= lim (1+ (x - 1))

=

2

1 x - 1

x 1

e

lim x e

:

f .

:

-

f .

:

N

g(x)1(1+ )g(x)

1 g(x)(1+ g(x)) .

:

f

:

h(x)

x

1 h(x)

x

1 lim (1+ ) = e,

h(x)

h(x)

lim (1+ h(x)) = e,

h(x) 0

049

• x

f x

x - 1 x - 2 : f(x) - 1

x + 1 x + 2 : lim f(x).

x x

x x

x

x x

x x

x x

x x

x 1 1- 1- 1- 0x = lim = lim =

x 1 1 1+ 0+ 1+

x

x 2 2- 1- 1- 0x= lim = lim =

x 2 2 1+ 0+ 1+

x

, :

lim (f(x) - 1) = 1

x

x

x

x - 1lim 1

x + 1

x - 2lim 1

x + 2

lim

=

=

f(x) = 2

:

f .

:

-

f

f

x 0 .

:

N o

-

.

:

1. -

-

f

-

f -

.

2. -

-

.

3.

, -

,

-

.

:

-

, -

, -

:

(

1).

1 .

050

• { }

{ }

• x

2 2 xx 1 x +

N :

lnx x + elim lim

x - 1 x + e

0

0

2DLH x 1

x 1

2

+ x+

2 xDLH x

(lnx)' = lim =

(x - 1)'

1

x= lim 2x

1 1 = = =

2 12x

(x + e )' = lim

(x + e )

2x 1

x

2 xx +

lnxlim

x - 1

1

2

x + elim

x + e

+ x +

xx DLH

x

xx

+ x +

xx DLH

='

1+ e = lim =

2x + e

(1+ e )' = lim =

(2x + e )'

e = lim =

2 + e

x

xx

x

xx

(e )' = lim =

(2 + e )'

e = lim =

e

1

:

-

f .

:

-

f .

:

-

-

.

:

1. -

0

0

.

2. -

-

( -

) .

3. -

,

2 .

4. -

, .

052

• 2x 0

N :

x( - x) + - 2xlim .

x

0 = 1

x 0

2

x 0

lim [x( - x) + - 2x] = 0 ( - 0) + - 20 = - 2

lim x = 0.

- 2 0 ,

.

- 2 = 0 .

= 2

= 2

0

0

2 2x 0 DLH x 0

x 0

x 0

:

x( - x) + 2 - 2x x( - x) + 2 - 2x lim = lim =

x (x )'

- x + xx + 2x = lim

2x

lim[ - x + xx + 2x] = - 1+ x 0 +

[ ]'

x 0

2 0 = - 1

lim 2x = 0.

- 1 0 ,

.

- 1 = 0 .

= 2 = 1 :

x 0

= 1

x(1 - x) + 2 - 2lim

2x 0

0

0

x 0 DLH

x 0

[x(1- x) + 2 - 2x]'= lim =

(x )'

1- x + xx + 2x = lim =

2x[1- x + xx + 2x]'

= lim(

2

x

x

x 0

=2x)'

- 1+ 0 + 0 = lim = .

21

- 2

:

-

-

0 -

.

:

.

:

N -

0

0 De LHospital .

:

1. -

-

0 .

,

(

),

0 .

2. -

.

3. -

,

-

.

:

.

053

• x 2

: lim ( - 2x)x .

+

+

+

+

x

2

0

0

DLHx

2

x

2

x

- 2x = lim =

1

x

- 2x = lim =

x

( - 2x)' = lim =

(x)'

= lim

x

2

lim ( - 2x)x

+

+

+

2 2

2

x

2

2

2

- 2 =

1-

x

= lim 2x =

= 2 =

2

= 2 1 =

2

:

(f g) .

:

o -

f g .

:

N -

-

.

:

1. -

0 .

2.

-

,

-

.

3. -

-

( -

) .

4. -

,

3 .

5. -

, .

054

• x

x + N : lim (lnx - e ) .

x

xx

+

+ x

xx x DLH

x

xx x

E

lnx= lim e - 1 =

e

lnx = lim e lim - 1 =

e

(lnx)' = lim e lim - 1 =

(e )'

x

x + lim (lnx - e )

x

xx x

x

xx x

1

x = lim e lim - 1 =e

1 = lim e lim - 1 =

xe

1 = + ( - 1) =

+ 1 = +

(0 - 1) =

=

-

:

(f - g) .

:

f - g .

:

N -

-

.

:

1. -

- .

2.

f - g :

g f

f (1- ) - g (1- )f g

.

3. -

,

f, g -

.

4.

,

-

(-

) .

5. -

,

4 .

055

• + +

x x

x 0 x 0

1( )x

N :

lim x lim

2 2

(lnx)'lnx 11(0 ) '

x x lnx x

DLHx 0 x 0 x 0

0- 2x(- x)'- x 0

(x x)'x x

DLHx 0 x 0 x 0

= lim e = lim e = lim e =

= lim e lim e = lim e

+

x

x 0lim x

=

x-xx

0

ln(1/x) ln(1/x)1 (0 ) x ln 1/x xx

DLHx 0 x 0 x 0

(ln(1/x))'

(x)'

x 0 x 0

=

= e =

= lim e = lim e = lim e

= lim e = lim

+

x

x 0

1( )x

1

lim =

.

2

2 2

2

1 1x (- ) - x x 1 1

- - x x

x 0

x x x

1 0 0x x

x 0 x 0

e = lim e =

= lim e = lim e = e = e =

1.

:

g ( x ) f(x)

:

g ( x ) f(x)

:

N

g ( x ) f(x)

.

:

1. :

g ( x ) g ( x ) l n f ( x )

g ( x ) l n f ( x )

f(x) = e =

= e

2. :

0 0

g ( x ) g ( x ) l n f ( x )

x x x xlim f(x) = lim e

3. -

e .

4.

-

.

056

• x-1

40x

x t + tN lim dt.

t + x

x-1

40

x-1 x-1

4 40 0

x-1

4 40

x-1

40

2

4x x

2

4

x > 10 < t < x - 1

t + 1 < x

x t + t dt

t + x

|x| |t|+|t| x 1+ 1 dt dt =

x| t + x |

x + 1 x

x t + tdt

t + x

x -+ 1 dt = (x - 1) =

x x

x - 1 lim = li

x

1

x

m

2

4

x-1

40x

x= 0

x

x t + t lim dt = 0

t + x

x-1

40x

x t + tlim dt = 0

t + x

:

x

cf(x,t) dt .

:

:

N

x

cf(x,t) dt

x

.

:

1.

x x

c c| f(x,t) dt| |f(x,t)| dt

2. ,

-

, |f(x,t)|

-

x,

.

3.

x x

c c| f(x,t) dt| |f(x,t)| dt

x x lim |f(x)| = 0 lim f(x) = 0

057