Κεφάλαιο 1 - Όριο - Συνέχεια Συνάρτησης - Όριο...

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Θεωρία και Ασκήσεις για το Μέρος Β Ανάλυση Κεφάλαιο 1 - Όριο - Συνέχεια Συνάρτησης - Όριο συνάρτησης στο xoεR - Ιδιότητες ορίων Μαθηματικά Κατεύθυνσης Γ' Λυκείου

Transcript of Κεφάλαιο 1 - Όριο - Συνέχεια Συνάρτησης - Όριο...

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    O

    1) Rxo

    1.1) Rxo

    3

    32)(

    2

    x

    xxxf

    ),3()3,( A .

    1)(3

    )3)(1()(

    xxf

    x

    xxxf

    1 f .

    f ),4()4,( B .

    1

    x A .

    x 3,

    3, 3 . ,

    )(xf 4, 4

    , x .

    , 3 x ,

    4 )(xf .

    )(xf , x 3 4

    :

    4)(lim3

    xfx

    f ),(),( oo xx

    fC ( 2).

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    2

    ox x , l

    )(xf .

    )(xf , x ox l

    )(xf ox l :

    lxfoxx

    )(lim

    :

    i) ox f ( 3)

    ii) )(lim xfoxx

    f

    : ),(),( oo xxa ),( oxa ),( ox .

    iii) ox f , )( oxf .

    )(lim xfoxx

    )( oxf . ( 4

    5).

    3

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    4

    5

    3,

    3,

    5

    3)(

    2

    x

    x

    x

    xxf . 6

    f .

    x 3 ( x

    3, )3,(x ) )(xf 6.

    x 3 ( x 3,

    ),3( x ) )(xf 2.

    )(xf , x 3

    6 6)(lim3

    xfx

    .

    )(xf , x 3

    2 2)(lim3

    xfx

    .

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    6

    f

    ),(),( oo xxa .

    i) )(xf 1l x

    ox , ),( oxa , )(xf , x ox

    1l 1)(lim lxfoxx

    .

    ii) )(xf 2l x

    ox , ),( oxx , )(xf , x ox

    2l 2)(lim lxfoxx

    .

    ( 7)

    y

    x

    O

    y'

    x

    x

    f(x) 1 3 5

    2

    -3

    6

    f(x)

    x

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    7

    To 1)(lim lxfoxx

    f ox ,

    2)(lim lxfoxx

    f ox .

    1l 2l f ox .

    :

    lxfxflxfooo xxxxxx

    )(lim)(lim)(lim

    y

    x

    O

    x

    f(x)

    xo x

    l2 f(x)

    x

    l1

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    1.2*) f Rxo

    .

    f ),(),( oo xxa .

    f ox Rl

    ( )

    ),(),( oo xxax ),(),( oooo xxxxx :

    ),()( llxf

    ( 8)

    8

    ),(),( oooo xxxxx

    ),( oo xxx oxx oxx0

    ),()( llxf lxf )(

    :

    f ),(),( oo xxa .

    y

    x

    O

    x

    l-

    xo

    l f(x)

    x xo- xo+

    l+

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    f ox Rl lxfoxx

    )(lim ,

    0 0 ),(),( oo xxax

    oxx0 lxf )( .

    * f ox .

    f ox , , .

    :

    i) 0)(lim)(lim

    lxflxfoo xxxx

    ii) lhxflxf ohxx o

    )(lim)(lim0

    :

    f ),(),( oo xxa .

    :

    lxfxflxfooo xxxxxx

    )(lim)(lim)(lim

    , )(lim)(lim xfxfoo xxxx

    , ox f .

    f ),( oxa .

    :

    )(lim)(lim xfxfoo xxxx

    f ),( ox .

    :

    )(lim)(lim xfxfoo xxxx

    , :

    f ox . :

    i) f

    ),(),( oo xxa .

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    ii) f ),( oxa

    . ( f

    ),( ox ).

    iii) f ),( ox

    . ( f

    ),( oxa ).

    xxf )( ( ) cxg )( , Rc

    ( ) R . :

    oxx

    xxfo

    )(lim oxx

    xxo

    lim

    cxgoxx

    )(lim ccoxx

    lim

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    1) f ),(),( oo xxa

    3)(lim 3

    xfoxx

    23)(lim xfoxx

    .

    R )(lim xfoxx

    .

    :

    )(lim xfoxx

    :

    113010103

    0)1)(1)(3(0)1)(3(0)3()3(

    03333)(lim)(lim

    22

    2323

    xfxfoo xxxx

    2) x

    xxxf 2

    2

    1)( .

    ) fC .

    ) fC : )(lim0

    xfx

    )(lim0

    xfx

    .

    ) , , )(lim0

    xfx

    .

    :

    x

    xxxf 2

    2

    1)( , *Rx :

    0,

    0,

    12

    1

    12

    1

    )(2

    2

    x

    x

    x

    xxf

    ) f .

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    ) :

    1)(lim0

    xfx

    1)(lim0

    xfx

    ) )(lim)(lim00

    xfxfxx

    , f , x 0.

    3) 3

    96)2()(

    2

    x

    xxxxf .

    ) f .

    ) , , )(lim3

    xfx

    ( fC ).

    :

    3

    96)2()(

    2

    x

    xxxxf , ),3()3,( x

    :

    y

    x

    O

    y'

    x

    -1

    2

    1

    3

    -2

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    3,

    3,

    2

    2)(

    ),3()3,(,3

    3)2()(

    ),3()3,(,3

    )3()2()(

    2

    x

    x

    x

    xxf

    xx

    xxxf

    xx

    xxxf

    ) f .

    ) :

    1)(lim3

    xfx

    1)(lim3

    xfx

    )(lim)(lim33

    xfxfxx

    , f , x 3.

    y

    x

    O

    y'

    x

    -1

    2

    1

    2

    3

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    1.3)

    ():

    1)

    i. 0)(lim

    xfoxx

    , 0)( xf ox . ( 1)

    ii. 0)(lim

    xfoxx

    , 0)( xf ox . ( 1)

    1

    y

    x

    O

    x

    xo

    l

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    1

    2) f g ox

    )()( xgxf ox . )(lim)(lim xgxfoo xxxx

    ( 2 2).

    2

    y

    x O x xo

    l

    y

    y

    x

    O

    x

    xo

    y

    Cf

    Cg

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    2

    (1) (2)

    ():

    f g ox .

    )

    i. 0)( xf ox , 0)(lim

    xfoxx

    .

    ii. 0)( xf ox , 0)(lim

    xfoxx

    .

    )

    i. )(lim)(lim xgxfoo xxxx

    , )()( xgxf ox .

    ii. )(lim)(lim xgxfoo xxxx

    , )()( xgxf ox .

    ) )()( xgxf ox , )(lim)(lim xgxfoo xxxx

    .

    3) f g : lxfoxx

    )(lim

    mxgoxx

    )(lim . :

    y

    x

    O

    x

    xo

    y

    Cf

    Cg

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    i. mlxgxfxgxfooo xxxxxx

    )(lim)(lim)()(lim

    ii. lcxfcxcfoo xxxx

    )(lim)(lim , Rc

    iii. mlxgxfxgxfooo xxxxxx

    )(lim)(lim)()(lim

    iv. 0)(lim

    xgoxx

    m

    l

    xg

    xf

    xg

    xf

    o

    o

    o

    xx

    xx

    xx

    )(lim

    )(lim

    )(

    )(lim

    v. lxfxfoo xxxx

    )(lim)(lim

    vi. kkxx

    k

    xxlxfxf

    oo

    )(lim)(lim ( 0)( xf

    ox ).

    vii. vv

    xx

    v

    xxlxfxf

    oo

    )(lim)(lim , *Nv

    : vov

    xxxx

    o

    lim .

    4) : ov

    v

    v

    v axaxaxaxP

    1

    1

    1 ...)( , 0va

    ox . : )()(lim oxx

    xPxPo

    :

    :

    )(...

    lim)(lim...)(lim)(lim

    lim)(lim...)(lim)(lim

    )...(lim)(lim

    1

    1

    1

    1

    1

    1

    1

    1

    1

    1

    1

    1

    ooo

    v

    ov

    v

    ov

    oxxxx

    v

    xxv

    v

    xxv

    oxxxx

    v

    vxx

    v

    vxx

    o

    v

    v

    v

    vxxxx

    xPaxaxaxa

    axaxaxa

    axaxaxa

    axaxaxaxP

    oooo

    oooo

    oo

    5) )(xP )(xQ x Rxo 0)( oxQ ,

    :

    )(

    )(

    )(

    )(lim

    o

    o

    xx xQ

    xP

    xQ

    xP

    o

    .

    :

    :

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    )(

    )(

    )(lim

    )(lim

    )(

    )(lim

    o

    o

    xx

    xx

    xx xQ

    xP

    xQ

    xP

    xQ

    xP

    o

    o

    o

    1) :

    i) 234

    23

    1 45

    34lim

    xxx

    xxx

    x

    , ii)

    82

    83lim

    2

    2

    2

    xx

    xx

    x

    :

    i) 234

    23

    45

    34)(

    xxx

    xxxxf

    1x

    .

    (3iv) 1ox . :

    xx

    x

    xx

    x

    xxx

    xxx

    xxx

    xxx

    xxx

    xxxxf

    4

    3

    )4(

    3

    )4)(1(

    )3)(1(

    )45(

    )34(

    45

    34)(

    2222

    2

    234

    23

    3

    2

    3

    2

    141

    31

    4

    3lim)(lim

    2211

    xx

    xxf

    xx.

    ii) 2x :

    82

    83)(

    2

    2

    xx

    xxxf .

    (3iv) 2ox . :

    )83(

    )2(

    )4(

    2

    )83)(4(

    )2(2

    )83)(4)(2(

    )2)(2(2

    )83)(4)(2(

    82

    )83)(4)(2(

    83

    )83)(82(

    )83)(83(

    82

    83)(

    2

    222

    2

    2

    22

    22

    22

    2

    2

    xx

    x

    x

    xxx

    x

    xxxx

    xx

    xxxx

    x

    xxxx

    xx

    xxxx

    xxxx

    xx

    xxxf

    :

    3

    1

    4

    4

    6

    2

    )2823(

    )22(

    )42(

    2

    )83(

    )2(lim

    )4(

    2lim)(lim

    22222

    xx

    x

    xxf

    xxx

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    2) , , ox :

    i) 4,

    4,

    1

    27

    6

    )(

    x

    x

    x

    xx

    x

    xf , 4ox

    ii) 2

    8)(

    2

    x

    xxxg , 4ox

    iii) 4

    168)(

    2

    x

    xxxh , 4ox

    :

    i) :

    3

    2

    74

    64

    7

    6lim)(lim

    44

    x

    xxf

    xx

    3

    2

    14

    24

    1

    2lim)(lim

    44

    x

    xxf

    xx

    3

    2)(lim)(lim

    44

    xfxf

    xx :

    3

    2)(lim

    4

    xf

    x

    ii) ),2()2,0[ gA :

    )42(2

    )42)(2(

    2

    22)2(

    2

    8

    2

    8

    2

    8)(

    223

    42

    xxxx

    xxxx

    x

    xxxx

    x

    xx

    x

    xx

    x

    xxxg

    : 842)4424(4)42(limlim)(lim444

    xxxxgxxx

    iii) :

    4,

    4,

    1

    1

    4

    4

    4

    )4(

    4

    168)(

    22

    x

    x

    x

    x

    x

    x

    x

    xxxh

    1)(lim4

    xhx

    1)(lim4

    xhx

    )(xh ,

    x 4.

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    1.4)

    ,

    .

    gf , h :

    i. )()()( xhxgxf ox .

    ii. lxhxfoo xxxx

    )(lim)(lim

    : lxgoxx

    )(lim

    4

    5)4()(

    2

    2

    xxxf

    ),2()2,2()2,( A . )(lim2

    xfx

    .

    :

    Ax :

    )4(4

    5)4(

    4

    5)4()( 2

    2

    2

    2

    2

    xx

    xx

    xxf

    Ax :

    )4()()4()4()( 222 xxfxxxf

    0)4(lim 22

    xx

    0)4(lim 22

    xx

    0)(lim2

    xfx

    (

    ).

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    1.5)

    :

    Rx xx .

    ( 0x , 0x

    ).

    ( ).

    :

    ) oxx

    xxo

    lim

    ) oxx

    xxo

    lim

    ( ).

    :

    )

    i. 1lim0

    x

    x

    x

    ii. 1lim0

    ax

    ax

    x

    , ( *Ra )

    )

    i. 01

    lim0

    x

    x

    x

    ii. 01

    lim0

    x

    x

    x

    , ( *Ra )

    )ii) )ii) )i) )i) ,

    uax ).

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    1.6)

    ))((lim))((lim xgfxfogoo xxxx

    .

    uxg )( oxx

    uxgo

    )(lim ( ).

    lufouu

    )(lim ( ).

    , ouxg )( ox , : )(lim))((lim ufxgfoo uuxx

    ))((lim xfogoxx

    ouxg )( ox .

    :

    i) x

    x

    x 3

    5lim

    0

    ii) 393

    3lim

    0 x

    x

    x

    iii) x

    x

    x 2

    1lim

    :

    i) : xx

    x

    x

    x

    xx

    x

    x

    x

    53

    5

    3

    3

    5

    5

    35

    5

    3

    5

    ux 5 vx 3 05limlim00

    xuxu

    03limlim00

    xvxv

    .

    :

    3

    5

    03

    5

    1

    11

    3

    5lim

    1limlim

    3

    5limlimlim

    53

    5lim

    3

    3lim

    5

    5lim

    53

    5

    3

    3

    5

    5lim

    3

    5lim

    000000

    00000

    u

    v

    vu

    u

    uv

    v

    u

    u

    xx

    x

    x

    x

    xx

    x

    x

    x

    x

    x

    uvuuvu

    xxxxx

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    ii) :

    x

    xx

    x

    xx

    xx

    xx

    x

    x

    3

    )393(3

    993

    )393(3

    )393)(393(

    )393(3

    393

    3

    ux 3 03limlim00

    xuxu

    .

    :

    61)39(

    lim)393(lim3

    3lim)393(lim

    393

    3lim

    00000

    u

    ux

    x

    xx

    x

    x

    uxxxx

    iii) :

    )1(

    1

    )1)(1(

    1

    1

    1122 xxx

    x

    x

    x

    x

    x

    :

    2

    1

    )1(

    1

    )1(

    1lim

    1lim

    2

    xx

    x

    xx

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    1) f R

    53)(2 xxf Rx . )(lim5

    xfx

    .

    :

    Rx :

    2

    53)(

    2

    53

    53)(253

    53)(2553)(2

    xxf

    x

    xxfx

    xxfxxxf

    : 2

    3

    2

    53lim

    5

    x

    x

    2

    3

    2

    53lim

    5

    x

    x ( )

    2

    3)(lim

    5

    xf

    x.

    2) x

    vxxx

    x

    ...2lim

    0.

    :

    :

    vx

    vxv

    x

    x

    x

    x

    x

    vxxx

    ...

    2

    22

    ...2.

    :

    2

    )1(...211...121

    lim...2

    2lim2lim

    ...2lim

    0000

    vvvv

    vx

    vxv

    x

    x

    x

    x

    x

    vxxx

    xxxx

    ( 11 a 1 ).

    3) : 1,

    1,

    9)42(3

    )14(9)(

    232

    x

    x

    xa

    xxaxf

    .

    ,a )(lim1

    xfx

    .

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    :

    :

    149))14(9(lim)(lim 2223211

    axxaxfxx

    91269)42(39)42(3lim)(lim11

    aaxaxfxx

    )(lim1

    xfx

    :

    2

    3

    3

    1032013

    0)32()13(0)9124()169(

    9126149)(lim)(lim

    2222

    22

    11

    aa

    aaa

    aaxfxfxx

    4) 6

    62)(

    x

    axxf . Ra ,

    Rxfx

    )(lim6

    .

    :

    Rlxfx

    )(lim6

    : )()6(626

    62)( xfxax

    x

    axxf

    . :

    243612

    6120612)]()6[(lim62lim66

    aa

    alaxfxaxxx

    5) f 6)(lim3

    xfx

    6)(

    819)(3)()(

    xf

    xfxfxg . )(lim

    3xg

    x.

    :

    :

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    519)(1

    33)(

    1

    519)()6)((

    2519)(

    33)()6)((

    93)(

    519)()6)((

    519)(519)(

    33)()6)((

    33)(33)(

    6)(

    519)(

    6)(

    33)(

    6)(

    819)(3)()(

    xfxf

    xfxf

    xf

    xfxf

    xf

    xfxf

    xfxf

    xfxf

    xfxf

    xf

    xf

    xf

    xf

    xf

    xfxfxg

    :

    15

    4

    60

    16

    10

    1

    6

    1

    5196

    1

    336

    1

    519)(

    1lim

    33)(

    1lim)(lim

    333

    xfxf

    xgxxx

    6) f R :

    i) Rlxfx

    )(lim2

    ii) )2()()2( xxfx Rx

    l .

    :

    Rx )2()()2( xxfx .

    20)2( xx )2(

    )2()(

    x

    xxf

    )2(

    )2(lim)(lim

    22

    x

    xxf

    xx

    . 1lim

    )2(

    )2(lim

    02

    u

    u

    x

    x

    ux

    ( 2 xu ).

    : 1)(lim2

    xfx

    (1)

    20)2( xx )2(

    )2()(

    x

    xxf

    1lim)2(

    )2(lim)(lim

    022

    u

    u

    x

    xxf

    uxx

    . ( 2 xu ). : 1)(lim

    2

    xf

    x (2)

    (1) (2) : 1)(lim2

    xfx

    1l .

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    7) 6

    3525lim

    3

    6

    x

    xx

    x.

    :

    ux 6 5 :

    3336 55 uxux ,

    23226 55 uxux

    555 66666 uxuxux

    :

    6

    7

    1

    33lim

    )1)(1(

    )33)(1(lim

    )1)(1(

    )1)(1(2)1)(1(lim

    1

    )1(2)1(lim

    65

    32lim

    6

    3525lim

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    8) RRf : 0)(lim1

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  • 26 : - http://sciencephysics4all.weebly.com/

    sciencephysics4all.weebly.com

    9) 32

    5)2()(

    2

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  • 27 : - http://sciencephysics4all.weebly.com/

    sciencephysics4all.weebly.com

    02

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