Κατηγορίες ασκήσεων στα όρια

of 59 /59
  • Author

    -
  • Category

    Education

  • view

    6.684
  • download

    2

Embed Size (px)

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