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  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    x 0limln x 2 ln2 0

    3xx 0lim e 2x 1 0

    3xf x e 2x 1,x R

    f R 1 2

    x ,x R :

    1 23x 3x

    1 2 1 2x x 3x 3x e e

    1 2 1 2 1 2

    x x 2x 2x 2x 1 2x 1

    1 23x 3x1 2 1 2e 2x 1 e 2x 1 f x f x

    , x 0 f x f 0 0

    x 0

    1lim ln x 2

    f x

    x 0 f x f 0 0

    x 0

    1lim ln x 2

    f x

    .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    k ,

    3xx 0

    ln x 2lim k

    e 2x 1R

    x 0

    3x

    ln x 2g x

    e 2x 1

    3xg x e 2x 1 ln x 2 , x 0

    x 0 x 0

    3xg x e 2x 1 ln x 2 k 0 ln2,lim lim !

    ,

    3xx 0

    ln x 2lim

    e 2x 1

    3xx 0

    ln x 2lim

    e 2x 1

    0 ,

    3

    ln 20 1

    e 2 1

    12 03 3 0 3 0 e 1 e 2 1 0 ln 2 0 2 1 1,

    , 0

    ,

    3xx 0

    ln x 2lim

    e 2x 1

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    3xx 0

    ln x 2lim

    e 2x 1

    0 ,

    3

    ln 20 2

    e 2 1

    22 03 3 0 3 0 e 1 e 2 1 0 ln 2 0 2 1 1,

    , 0

    .

    )

    3xg x e 2x 1,x R

    1 2

    x ,x 1 2

    x x :

    1 2

    1 2

    3x 3x3x 3x1 2

    1 2 1 2

    1 2 1 2

    3x 3x e ee 2x 1 e 2x 1 g x g x

    2x 2x 2x 1 2x 1

    g R

    0g 0 e 0 1 0

    x 0 g x g 0 0 g x 0

    x 0 g x g 0 0 g x 0

    0 1 ,

    x 0limf x

    :

    x 0x 0 1

    x 0

    x 0

    1lim ln x

    lim f x g x ln ln 0

    ,!

    ln 0lim f x ln1lim ln x

    g x

    1:

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    0

    0

    3x 3x 3xx 0 x 0 x 0 x 0 x 03x

    1ln x 1ln x 1 1 1x 1limf x lim lim lim lim

    5e 2x 1 3e 2 x 1 3e 2e 2x 1

    0

    x 0limf x

    ) 1

    3x

    ln x 1f x ,x 1,x 0

    e 2x 1

    :

    lnx x 1, x 0 1 ( x 1)

    xe x 1, x 2R ( x 0 )

    1 x x 1 :

    ln x 1 x , x 0

    2 x 3x :

    3xe 3x 1, x 0

    x 0:

    3x 3x

    3x

    0 ln x 1 xln x 1 x ln x 1 x 1

    f x1 1 5e 3x 1 e 2x 1 5x 0 05xe 2x 1

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    :

    3x 3x 3xx 0 x 0 x 0 x 0

    ln(x 2) 1 1lim lim ln(x 2) lim ln(x 2) lim

    e 2x 1 e 2x 1 e 2x 1.

    :

    x 0

    lim ln(x 2) ln(0 2) ln2 0 .

    : 3xf(x) e 2x 1, x R .

    1 2

    x , x R 1 2

    x x , :

    1 23x 3x

    1 23x 3x e e

    1 22x 2x .

    ,

    1 2 1 23x 3x 3x 3x

    1 2 1 2 1 2e 2x e 2x e 2x 1 e 2x 1 f(x ) f(x ) ,

    f , 3 0f(0) e 2 0 1 0 .

    f 3x1

    f (x) e

    2

    f (x) 2x 1. , ( ,0) (0, ) .

    :

    ( ,0) (0, )

    : 0x 1 1

    0f(x ) 3

    3

    1e 2 1 3 0

    e 3 3e 2 1 e 1 0

    f

    , :

    3x 0x 0lim e 2x 1 e 1 0 .

    , ,

    3xx 0

    1lim

    e 2x 1

    3xx 0

    1lim

    e 2x 1.

    , :

    3xx 0

    ln(x 2)lim ln2 ( )

    e 2x 1

    3xx 0

    ln(x 2)lim ln2 ( )

    e 2x 1.

    , :

    3x 3xx 0 x 0

    ln(x 2) ln(x 2)lim lim

    e 2x 1 e 2x 1.

    ,

    3xx 0

    ln(x 2)lim

    e 2x 1 .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    T

    3ln(x 2)

    e 2x 1 2,0 (0, ) 3f(x) e 2x 1

    3f (x) 3e 2 0

    f < R. x 0 f(x) f(0) 0 x 0 f(x) f(0) 0 .

    .

    3 3xx 0 x 0

    1 ln(x 2)lim lim

    e 2x 1 e 2x 1

    3x 3xx 0 x 0

    1 ln(x 2)lim lim

    e 2x 1 e 2x 1

    x 0lim(ln(x 2)) ln2 0 .

    .

    ) 1

    x 0limln x ln 0

    3x 3xx 0 x 0

    1 ln(x )lim lim

    e 2x 1 e 2x 1.

    3x 3xx 0 x 0

    1 ln(x )lim lim

    e 2x 1 e 2x 1.

    .

    0 1

    x 0limln x ln 0

    3xx 0

    ln(x )lim

    e 2x 1,

    3xx 0

    ln(x )lim

    e 2x 1.

    .

    1 .. 0

    0 de L H.

    3xx 0

    1 1lim

    5(3e 2)(x 1).

    . 0 .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    ) ( 3xe + 2x -1 > 0 x > 0 )

    3x 3x

    3x

    ln(x 1) 1e 2x 1 5ln(x 1) e 2x 5ln(x 1) 1

    5e 2x 1.

    3x(x) e 2x 5ln(x 1) x 0

    (0) 1

    3x 5 (x) 3e 2x 1

    (0) 0

    3x

    2

    5 (x) 9e 0

    (x 1).

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    3xf x e 2x 1 , x R

    f :

    3xf x 3e 2 0 x R

    R f x 0 x R f

    R

    :

    f

    x 0 f x f 0 f x 01

    f

    x 0 f x f 0 f x 0 1

    x 0 :

    3xx 0 x 0

    ln x 2 1lim lim ln x 2

    f xe 2x 1

    x 0lim f x 0 f x 0 x 0

    x 0

    1lim

    f x

    x 0limln x 2 ln2 0

    x 0 :

    3xx 0 x 0

    ln x 2 1lim lim ln x 2

    f xe 2x 1

    x 0lim f x 0 f x 0 x 0

    x 0

    1lim

    f x

    x 0limln x 2 ln2 0

    E .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    3xh x e 2x 1 0

    x 0 :

    3x 3x

    ln x 2ln x 2 x

    e 2x 1 e 2x 1

    x

    1

    3x

    x 0

    e 2x 1lim

    x

    0

    0

    3x

    3x

    x 0 x 0

    e 2x 1lim lim 3e 2 5

    x

    DLH :

    3x

    x 0

    e 2x 1lim 5

    x

    x 0 x 0

    ln x 2 1lim lim ln x 2

    x x

    1 :

    3x 3xx 0 x 0

    ln x 2ln x 2 xlim lim

    e 2x 1 e 2x 1

    x

    x 0 :

    3x 3x

    ln x 2ln x 2 x

    e 2x 1 e 2x 1

    x

    1

    3x

    x 0

    e 2x 1lim

    x

    0

    0

    3x

    3x

    x 0 x 0

    e 2x 1lim lim 3e 2 5

    x

    DLH :

    3x

    x 0

    e 2x 1lim 5

    x

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    x 0 x 0

    ln x 2 1lim lim ln x 2

    x x

    1 :

    3x 3xx 0 x 0

    ln x 2ln x 2 xlim lim

    e 2x 1 e 2x 1

    x

    0 0

    3x

    ln x 2

    e 2x 1 .

    http://lisari.blogspot.gr/

  • ___________________________________________________________________________ 6 http://lisari.blogspot.gr . 2016-17

    3xh(x) e 2x 1 , x .R h < R .

    x 0 3xh(x) 0 e 2x 1 0 (1) , x 0 3xe 2x 1 0 (2) .

    (x) = ln(x 2) , x 2 2,

    .

    3x

    x 0

    ln(x 2)lim

    e 2x 1=

    3x

    x 0

    1lim ln(x 2)

    e 2x 1=

    x 0limln(x 2) ln2 0 ,

    3x

    x 0

    1lim

    e 2x 1 (1).

    3x

    x 0

    ln(x 2)lim

    e 2x 1=

    3x

    x 0

    1lim ln(x 2)

    e 2x 1= ,

    x 0limln(x 2) ln2 0 ,

    3x

    x 0

    1lim

    e 2x 1 (2).

    3xx 0

    ln(x 2)lim

    e 2x 1 .

    ) 3xh(x) e 2x 1 , x .R 3xh (x) 3e 2 0 h < R .

    x 0 3xh(x) 0 e 2x 1 0 (1) , x 0 3xe 2x 1 0 (2) .

    (x) = ln(x ) , x ,

    .

    = 1

    x 0limf(x)=

    3xx 0

    ln(x 1)lim( )

    e 2x 1

    0

    0

    3xx 0

    11x 1lim( )53e 2

    .

    > 1

    x 0lim f(x) =

    3x

    x 0

    ln(x )lim

    e 2x 1=+ (1).

    x 0lim f(x) =

    3x

    x 0

    ln(x )lim

    e 2x 1=- (2).

    http://lisari.blogspot.gr/

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