___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

M

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

) 2xe 0 x R :

2x 3 2x 32x

3 3

e 1 e f x lim f x lim f x e

2016 x R

) f R g

f , g

g R

x x

g lim g x , lim g x ,R R

0 g R 0x R : 0g x 0

g 1 1 0

x

)

g

0 0 0 0 0

1 1 1x x g x g x g x 0

2016 2016 2016

1

1

lnx x 1 x 0 ( x 1 )

, :

)

)

) :

7 : (30/10/2016)

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

2xe 0 x R , x 2xe :

2x 2x 2xlne e 1 e 2x 1 x R

2xe 1 x 0

x g x : 2g xe 2g x 1 0 x R

0g x 0 x x

0

x x :

2g x 2g x

1 1g x g x

2016 2016 1 1g x 2

2016f g x 2g x 1 e 2g x 1 e 2g x 1

1g x

2016 2g xe 2g x 1 0x ,

00x x

1 1limg x g x 0

2016 2016

0

0

2g x 2g x

0x xlim e 2g x 1 e 2g x 1 0

2g xe 2g x 1 0 0x ,:

02g xx x

1lim

e 2g x 1

:

02g xx x

1 1lim g x

2016 e 2g x 1

2 :

0 02g xx x x x

1g x

2016 1 1lim lim g x

2016f g x 2g x 1 e 2g x 1

:

2

0 02g xx x x x

1g x

2016 1 1lim lim g x

2016f g x 2g x 1 e 2g x 1

.

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

,

:

2f x x 1

x 1

f x f 1f 1 lim ... 2

x 1

, :

x 1

f x f 1lim ... 2

x 1

x 1

f x f 1lim

x 1R f 1 2

.

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

) x R 2xe 0

2x 32x

3

e f(x) lim e

.

) f < R ( < ) g < R

,

R . , 0

x

)

00x x

1 1lim x x

2016 2016

1u x

2016

0

01u x

2016

1lim g u g(x )

2016

g

00x x

1 1limg x g x

2016 2016

g

0 0 0 0

1 1x x g x g x 0

2016 2016

1

t t 1s , 7

1 cm / s12

.

) 1

t 0 ,

.

:

i 0t i1

t s2

, , i

t 0 .

.

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

x 0 :

3 3 32 2 2 2 2 2

4 4 4 2 2 2

3x 4 8 4x 3x 4 2 8 4x 2 1 3x 4 2 8 4x 2

x x x x x x

2u x u 0 x 0 :

x 0 u 0 u 0

x 0 u 0 u 0

2

2

3 2 3

2

f u f 03x 4 2 3u 4 2 3f 0

u u 0 4x

g u g 08 4x 2 8 4u 2 2g 0

u u 0 3x

lim lim lim

lim lim lim

:

f u 3u 4

3f u

2 3u 4

3f 0

4

3 2g u 8 4x

2

3

4g u

3 8 4u

1g 0

3

:

x 0 x 0 x 0

3 32 2 2 2

4 2 2 2

3x 4 8 4x 1 3x 4 2 8 4x 2 3 1

4 3x x x xlim lim lim

1 0 (20/11/2016)

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

x 0 :

k 2 k m 4 m 4 m 4k k2 k 2 km m m

6 6 6 4 2 2 4

x x x x x 1 x 1

x x x x x x x

2u x u 0 x 0 :

x 0 u 0 u 0

x 0 u 0 u 0

k k2 k k

2 k 1

m 4 mm m

4 m 1

f u f 0x u f 0

u u 0x k

g u g 0 x u g 0

u u 0x m

lim lim lim

lim lim lim

:

k kf u u

11

k k

f u u k

k 1

f 0k

mmg u u

11

m m

g u um

m 1

g 0m

:

x 0 x 0 x 0 x 0 x 0

k 2 k m 4 m 4k 2 km m

6 4 2 2 4

,,,m,k 0

k 1 m 1

x x x 1 x 1

x x x x x

k m

lim lim lim lim lim

:

:

k,m,, 2,3,... ,, 0 :

x 0

k k mm

, ,

x x

, x

,

lim

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

0

0 2 x 2 , x 0

3 32 2 2 2

4 4 4

2 2

34 2 4 2 2 23

32 2 2 2 2 23

3x 4 8 4x 3x 4 2 8 4x 2f(x)

x x x

3x 4x

x 3x 4 2 x (8 4x ) 2 8 4x 8

3 4

x 3x 4 2 x (8 4x ) 2 8 4x 8

2x 0

1lim

x

2x 0

3 3lim

43x 4 2,

32 2 2x 0 3

4 1lim

4(8 4x ) 2 8 4x 8

x 0limf(x) .

0

0

m m

4 4

x

, x 0

k 2 k m 4 m 4k 2 km m

6 6 6

x x x x f(x)

x x x

42

k 1 m 16 2 k k 1 6 m 4 m 1k m

4 2k 1 m 12 k k 1 m 4 m 1k m

xx

x x ... x x ...

1 1

x xx ... x ...

:

k 1 m 1x 01 1

limf(x)k m

,

.

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

:

32 2

4

3x 4 8 4x

x=

2

4

3x 4 2

x -

3 2

4

8 4x 2

x=

=2

4

3x

x

2

1

3x 4 2+

2

4

4x

x

2

2 23 3

1

8 4x 2 8 4x 4

=

=2

3

x

2

1

3x 4 2+

2

4

x

2

2 23 3

1

8 4x 2 8 4x 4

2 2x 0

3 1lim

x 3x 4 2= ,

2 3 2x 0

4 1lim

x 8 4x 2=

x 0lim

32 2

4

3x 4 8 4x

x=

x 0 ,

k 2 k m 4m

6

x x

x=

k 2 k

6

ax

x -

m 4m

6

x

x=

2

6

x

x

k 1 k 2

2 k 2 k 2 2 k k 3 k 2 2 k k 1kk k

1

x x (x ) ..... x

+

4

6

x

x

m 1 m 2 m 3

m 4 m 4 2 m 4 m 2 m 4 m 1mm m m

1

x x x ..... x

=

4

x

k 1 k 2

2 k 2 k 2 2 k k 3 k 2 2 k k 1kk k

1

x x (x ) ..... x

+

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

2

x

m 1 m 2 m 3

m 4 m 4 2 m 4 m 2 m 4 m 1mm m m

1

x x x ..... x

x 0lim (

4

x

k 1 k 2

2 k 2 k 2 2 k k 3 k 2 2 k k 1kk k

1

x x (x ) ..... x

)= ,

x 0lim (

2

x

m 1 m 2 m 3

m 4 m 4 2 m 4 m 2 m 4 m 1mm m m

1

x x x ..... x

)= ,

:

x 0lim

k 2 k m 4m

6

x x

x=

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___________________________________________________________________________ 2016 http://lisari.blogspot.gr . 2016-17

:

232 2

32 2

4x 0 x 0 34 2 2

3x 4 8 4x3x 4 8 4xlim lim

x x 3x 4 8 4x

3 22 2

2x 0 2 223 3 34 2 2 2 2 2 2

3x 4 8 4xlim

x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

6 4 2

x 0

27x 108x 144x 64lim

4 216x 64x 64

22 22

3 3 34 2 2 2 2 2 2x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

2 4 2

2x 0 2 223 3 34 2 2 2 2 2 2

x 27x 92x 208lim

x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

4 2

2x 0 2 223 3 32 2 2 2 2 2 2

27x 92x 208lim

x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

4 2

x 0

2x 0 2 223 3 32 2 2 2 2 2 2

lim 27x 92x 208

1lim

x 3x 4 8 4x 3x 4 3x 4 8 4x 8 4x

208 ,

22 22

3 3 32 2 2 2 2 2 2