01limites.continuite_2 by RedaNachi

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									                                                                                                                                                          1
                                                                                                      :               ‫تا‬                ‫ا‬                 ‫أ‬
                                                                    lim ( 9 x + x + 1 + 3x) (2 lim ( 9 x 2 + x + 1 − 2 x) (1
                                                                                     2
                                                                    x → −∞                                                 x → +∞


                                                                   lim ( 1 − 2 x − − x + x + 1)(4
                                                                                         3                3
                                                                                                                                      lim ( x 2 − x + 2 )(3
                                                                   x →−∞                                                          x → +∞

         x2 − 4            x2 − 4                   1 − 3x − 2                               x +1+ − x +1                                       x − x2 + x − 1
lim−            (9 lim+           (8       lim                 (7                 lim                                 (6          lim                              (5
x → −2   x+2       x→2     x−2             x → −1     x +1                       x → −∞
                                                                                             x2 − x2 + 2                          x → +∞
                                                                                                                                                 x2 − x4 − 1
                                         x+ x                              2+ x + 3− x −3           x 2 − x − 6 + 3x − x 2
                                  lim                  (12     lim                        (11 lim                          (10
                                  x →0
                                         x 2 +x −x             x → −1           x +1          x →3−           x+3
                                                                                                                               1
                                                                                                              lim                         (13
                                                                                                              x →0    3 cos x − sin x − 3
                                                                                 tan x − 1          cos x − 3 sin x
                                                                    lim                    (15 lim                  (14
                                                                    x→
                                                                       π
                                                                               2 cos x − 2     x→
                                                                                                  π      6x − π
                                                                           4                      6

                                                                                                                                                          2
                                                                               1
                                                               ( x − 1) sin( x − 1) ; x < 1
                                                               
                                                      f ( x) =  x 2 − 2 x − 8                                                ‫ا ا‬
                                                                                    ; x ≥1
                                                                x−2 −2
                                                               
                                                      lim f ( x)                ‫ 2( ا‬x0=1                         f ‫ل‬             ‫ وادرس ا‬Df ‫د‬                     (1
                                                     x → +∞


                                                                   4             ‫ا‬               ‫ل‬                ‫ا‬                             f     ‫ا ا‬     ‫3( ه‬
                                                                                                                                                          3
                                                   x + 2x − a
                                                       2

                                                               ;x > 2
                                                       x−2
                                         f ( x) =  2                                                                         ‫ا‬             f          ‫ا ا‬
                                                   2 x + b − a ;x ≤ 2
                                                  
                                                        x

                                              2            ‫ا‬                         f               ‫نا ا‬                             b‫ و‬a            ‫د‬       ‫دا‬
                                                                                                                                                          4
                                                                                                     x − 4x + 4x
                                                                                                          3           2
                                                                                     f (x ) =                                                   ‫ا ا‬
                                                                                                        x 2 −4
                                                                        D f ‫ات‬                         ‫ت‬      ‫ا‬                               ‫ وأ‬D f ‫د‬ (1
                                                     ‫2 و 2- ؟‬              ‫آ‬                 ‫ل‬          ‫ا‬                                   f    ‫2( ه ا ا‬
                                                                                                                                                       5
                                                                                              1 + sin x − 1   2
                                                                           f (x ) =                                                   ‫ا ا‬
                                                                                                   x2
                                                                                                                                                   Df ‫د‬            (1
                                                                   .0            ‫ل‬                    ‫ا‬                           f             ‫أن ا ا‬             (2
                                                                                                                                                       6
                                   3 cos x − sin x      π
                          f   (x ) =               ;x ≠
                                     2 cos x − 1        3
                                                                                             :                                       ‫ ا‬f           ‫ا ا‬
                          f    π   2
                               ( )=
                          
                               3    3
                                                                                                                π
                                                                                                            .                           f ‫أن‬
                                                                                                                3
                                                                                                                                        7
                                                                        2
                                                        f ( x ) = x sin   , x ≠ 0
                                                                        x         :f ‫ا ا‬
                                                         f ( 0) = 0
                                                        
                                                   . lim f ( x )       ‫2( ا‬            .0                                   f‫ل‬          ‫1( ادرس ا‬
                                                     +∞

                                                                                                             8
                                 IR                       ‫ا‬                   x + x − x + x+1= 0 ‫د‬
                                                                               5           3       2
                                                                                                           ‫أن ا‬                                       (1
                                   IR          ‫ا‬    ‫و‬                    3 x + 2 x + x − 10 3 = 0 ‫د‬
                                                                            7     5         6 4
                                                                                                          ‫أن ا‬                                        (2
[0,1]   ‫ل‬       ‫ا‬               ‫را‬                 Cf                    ‫أن ا‬      . f (x ) = x + x − 1
                                                                                                4
                                                                                                        ‫ا ا‬                                           (3
                                                                        g ( x) = − x 3 ‫ و‬f ( x) = x + 1 :                          ‫ا ا‬                (4
         7       3
        − <α < −            :            α              ‫ة أ‬         ‫و‬                  ‫ن‬                   Cg ‫ و‬Cf                          ‫أن ا‬
         8       4
                                                                                                                                         9
                                                   f(1)=1‫ و‬f(0)=0                          [0,1]                   ‫ دا‬f
                                                                                                                   1− c
                                                                                           '( ∃c ∈ ]0,1[): f (c) =      ‫أن‬
                                                                                                                   1+ c
                                                                                                                                    10
                                                                                       [a , b ]                             ‫ دا‬f
                                                                                     (a < b )                       f (b ) > b 2 ‫ و‬f (a ) < ab
                                                        f (c ) = bc                [a , b ]   c                           ‫د‬           ‫أﻩ‬
                                                                                                                                     11
                          ∃k ∈ IR ; ∀( x, y ) ∈ IR ² : f ( x) − f ( y ) ≤ k × x − y
                                   *
                                   +                                                                                    ‫د‬          ‫ دا‬f
                                                                                                  IR                            ‫ دا‬f ‫أن‬
                                                                                                                                    12
                        . ( ∀x ∈[0,1]): f ( x) ≥ 0 ‫ و‬f(1)=f(0)=0                               [0,1]                            ‫ دا‬f
                                                                                                                      1
                                                                           ( ∀n ∈ IN * )( ∃c ∈[0,1]): f ( c) = f ( c + ) ‫أن‬
                                                                                                                                    n
                                                                                                                                    13
            +                      +                                               +                   +
(∀x ∈ IR ) : f ( x) < x ‫ و‬IR                                  f :             IR                  IR     ‫ دا‬f
                                                                                              . f (0) = 0 ‫أن‬                                     (1
                                             (∀(a, b) ∈ ( IR*+2 ))(∃M ∈ [0,1[)(∀x ∈ [a, b]) : f ( x) ≤ Mx : ‫أن‬                                   (2
                                                                                                                                   14
    . ]0,1[ ‫ل‬       ‫ا‬       an ‫ا‬        ‫و‬                 Arc cos( x) − x = 0 ‫د‬n
                                                                                              ‫ ا‬IN           n  *
                                                                                                                                   ‫أﻩ‬              (1
                                                                                                  1
                                                                                             .         ‫ و‬a n ‫رن ا د‬                                (2
                                                                                                  2
                                                                                   . (∀n ∈ IN * ) : a n +1 > a n : ‫أ ﻩ‬                             (3
                                                                                                                                    15
                                       . ∀x ∈ [a, b] : f ( x) > 0                      [a; b]                               ‫د‬       ‫ دا‬f
                                                                                               ∃m > 0, f ( x) ≥ m ‫أن‬                         ‫أ‬
                                                                                                                                        16
                                              125                 125
                            A= 3+ 9 +             et B = −3 + 9 +                                                       ‫د‬          ‫ا‬
                                               27                  27
                                                                                               3
                                                                                                    AB            ‫و‬       A −B                       ‫1( أ‬
                                                                 125 3         125
                                                  x = 3 3+ 9 +      − −3 + 9 +                                                     ‫د‬      ‫ا‬           (2
                                                                 27            27
                                                      . x = 1 ‫أن‬     ‫( ا‬b . x                                                      x3                ‫( أ‬a
                                                                                                                                        17
                                                                               :                ‫د تا‬                  ‫ ا‬IR
  1− 3 x 3              x 4 = −2 (4
(        ) + 8 = 0 (5                         x 6 = 6 (3 ( x + 1) 3 = −27 (2                                           ( 2 x − 1) 5 = 32 (1
  3− 3 x
                                                  (   t=   6
                                                               1+ x       ‫و‬                )    3
                                                                                                    1 + x − 3 1 − x = 6 1 − x 2 (6
                                                               1− x

                                                                                                                                        18
                                                                 f ( x ) = 2x − 4x + 1 2
                                                                                         ‫ا ا‬
                                                                        . ‫ه‬            ‫ وا‬f ‫ات‬                                           ‫1( ادرس‬
                                                           I = [1, +∞[ ‫ل‬      ‫ا‬      f ‫را ا‬                                             g     (2
                                                                J‫ل‬                 I‫ل‬                   ‫ا‬                          g ‫أن‬          (a
                                                                                               C g −1             ‫وار‬            g −1 ( x) ‫د‬     (b

                                                                                                                                        19
                                                                                                4x                        ‫ا ا‬
                                                                                       f ( x) = 2
                                                                                               x +1
                                f   −1
                                         ( x) ‫د‬                                    ‫ل‬                        [− 1,1]                           f ‫أن‬
                                                                                                                                        20
                                                                                           2+ 4−x                 2
                                                                          f (x ) =                                               ‫ا ا‬
                                                                                              x
                                                                                                                                       Df ‫د‬           (1
                                                                I = ]0, 2] ‫ل‬               ‫ا‬                 f        ‫را ا‬              g             (2
                                                                J‫ل‬                 I‫ل‬                   ‫ا‬                          g ‫أن‬          (a
                                                                                                                                 g −1 ( x) ‫د‬     (b
                                                                                                                                        21
                                                                          :                   ‫ا‬f ‫ا ا‬
                                                                      f ( x) = ( x + 1 − 1) 3


                                                                                  . f ‫ا ا‬              ‫د‬    (1
                                         J‫ل‬                    [− 1,+∞[ ‫ا ل‬                   f ‫أن ا ا‬   (a (2
                                                                                                                            −1
                                                                                   .J                x                f          ( x) ‫د‬         (b
                                                                                                                                        22
                                                  f ( x) = x − 3 x + 3 x
                                                                      3    2           3
                                                                                                                  ‫ا ا‬
                                                                                               .f           ‫ا ا‬                          ‫د‬            (1
                                                                           f ( x) = x                        ‫د‬          ‫ ا‬IR       +
                                                                                                                                                      (2
                                                               f ( x) = (3 x − 1) 3 + 1 ‫أن‬                                                      (a (3
                                               [0,+∞[ ‫ا ل‬                   ‫ ا‬f ‫أن ا ا‬                                                          (b
                                            J‫ل‬         [0,+∞[ ‫ا ل‬               f ‫أن ا ا‬                                                        (c
                                                                                                                            −1
                                                                                   .J                x                f          ( x) ‫د‬          (d
                                                                                                                                              23
                                                                                                                     ‫ا‬f       ‫ا ا‬
                                                                    f (x ) = −x − 3 3 (1 − x ) 2 + 3 3 1 − x + 1
                                                                                      . f ‫ا ا‬                    ‫د‬                                             (1
                                  −1
                              . f ( x) ‫د‬                                  ‫ل‬          ]−∞,1]                 f ‫أن‬                                               (2
                                                                              . f (x ) = 1               ‫د‬            ‫]1,∞−] ا‬                                 (3
                                                                                                                                              24
                                                                                                x
                                                                                      f ( x) =                                   ‫ا ا‬
                                                                                               1+ x
                                                                                                                       1
                                                       (∀x ∈ ]−1, +∞[) : f (x ) = x + 1 −                                  : ‫أن‬                    (a (1
                                                                                                                      x +1
                     f   −1
                              ( x) ‫د‬                             J‫ل‬                  ]−1, +∞[                                    f ‫أن‬                 (b
                                                                                     −1
                                                                                    f ( x) = f ( x) : ‫د‬                              ‫ا‬R                        (2
                                                                                                                                              25
                                                       : f (x ) = (4 − 3 x 2 )3 :‫ب‬                                    ‫ا‬f         ‫ا ا‬
                                                                         Df : f    ‫ا ا‬                                                             ‫د‬           (1
                                                                 I = [ 0,8] ‫ل‬                  ‫ا‬                 f        ‫را ا‬                g                (2
                                           −1
                                       g ( x) ‫د‬                                       J‫ل‬                             I                        g ‫أن‬
                                                                                      . f ( x) = x                   ‫د‬           ‫ ا‬R                           (3
                                                                                                                                              26
                                                            3
                                                                x 2 −1
                                                f (x ) =        3                                                    ‫ا‬f          ‫ا ا‬
                                                                  x

                                                                      J‫ل‬                    ]0, +∞[                                      f ‫أن‬                  (1
                                                                                   (∀x ∈ IR ): x − x 2 + 4 < 0 ‫أن‬                                              (2
                                                                                          .J             x                   f −1 ( x)            ‫د‬            (3
                                                                                f (x ) = 5                ‫د‬               ‫[∞+ ,0] ا‬                            (4
                                                                                                                                              27
                                                                                                    :                    ‫تا‬               ‫ا‬                ‫أ‬
lim ( 3 −x 3 + 2x 2 − x − 2x 2 + 1 )                   (3 lim ( 3 1 − x 3 + 2x ) (2                              lim (3 8 x 3 − x + 1 − x) (1
x →+∞                                                       x →−∞                                             x → +∞


        lim ( 3 −3x 3 − 1 + x          3
                                           3)     (6       lim ( 3 −8x 3 + x 2 + 1 + 2x )                (5          lim ( 3 8x 3 − 1 − 2x ) (4
    x →−∞                                                x →−∞                                                       x →+∞

                                                                      3
                                                                          x 2 +1 − 3 1− x                                            3
                                                                                                                                         x 2 +1 + x
                                                         lim                                                      (8         lim                               (7
                                                        x →−∞ 4
                                                                  − x + 4x + 1 − 2 − x
                                                                          3                                                  x →−∞ 3     1− x − x 2
                          3
                              (x + 2) 2                                        3
                                                                                   x 2 −4                                        3
                                                                                                                                     x +1 −1
                  lim−                           (11                lim−                            (10                   lim                              (9
                 x →−2         x +2                               x →−2            x +2                                   x →0         x
                                                                                                     3
                                                                                                         x 2 −1 + x 2 + x − 2
                                                                                           lim                                                             (12
                                                                                              −
                                                                                           x →1                x −1
                                                                                                             3
                                                                                                                 (x + 1) 2 + x 2 + x
                                                                                                   lim                                                     (13
                                                                                               x →−1−                  x +1
                                                                                                   28
                                                                          :            ‫و تا‬                     ‫ا‬             ‫أ‬
                                                                           1           2 π
                                                                  Arc tan( ) + Arc tan( ) = (1
                                                                           5           3     4
                                                                 1           1          1 π
                                                         A rc tan + A rc tan + A rc tan = (2
                                                                 2           5          8 4
                                                                 1           1          1
                                                         A rc tan + A rc tan − A rc tan = 0 (3
                                                                 3           7          2
                                                                                      1    π
                                                  (∀x > 0) : A rc tan( x ) + A rc tan( ) =   (4
                                                                                       x   2
                                                                                     1     π
                                                (∀x < 0) : A rc tan( x ) + A rc tan( ) = − (5
                                                                                    x      2
                                                                                                        1
                                                (∀x ∈ IR ) : cos(Arc tan x ) =                                               (6
                                                                                                   1+ x 2
                           1 π                                       1          1   π
                0 ≤ arctan( ) ≤ ‫أن‬               )           4 arctan − arctan    =                                          (7
                           5   8                                     5         239 4
                                                                                                       1
                                   (∀x > 0) : Arc tan(x + 1) − Arc tan x = Arc tan(                         )                (8
                                                                                                  x 2 + x +1
                                                                                                   29

                                                         arctan 2 + arctan 3                                ‫أ‬
                                                                                                   30
                                                                 :      ‫ ا‬IR
                                                               x −8                   π
                                                      Arc tan(      ) − Arc tan( x) =
                                                                 8                    2
                                                                                 31
                                                        .        ‫د تا‬       ‫ ا‬IR
                x 2 −1                     π                                          π
    A rc tan(       2
                       ) + A rc tan( x ) =   (2            Arc tan 2x + Arc tan(3x ) =                                       (1
                  x                        2                                                                    4
                                                                                                   32
                                                                                 ‫د‬         ‫ ا‬IR
                                                                                                                             π
                                    (E ) : arctan(x − 1) + arctan x + arctan(x + 1) =
                                                                                                                             2
. ]0,1[   ‫إ‬                 ‫ وأن ه ا ا‬IR             ‫ا‬       ‫و‬                    (E )        ‫د‬             ‫ا‬                (1
                                                                                . (E )        ‫د‬             ‫ا‬                (2
                                                                                                   33
                                                                     :                 ‫ت‬         ‫ا‬       ‫أ‬
                                                             π
                                              Arc tan x −                                       arctan x
                                      lim              6             (2
                                                                                           lim             (1
                                                                                           x →0    x
                                      x→
                                          3          3
                                         3    x−
                                                    3
                                                    1 π
                                            Arc tan −                                               x
                                       lim−         x 2              (4         lim A rc tan(                       )        (3
                                       x →0       x                             x →1              x −1
                                                                                                    2


                                       1 π                                                              π
                        lim− x (arctan( ) + )                (6 .             lim x (Arc tan x − )                      (5
                        x →0           x   2                              x →+∞                 2
                                                                                  x +1 π
                                                     lim (x arctan(                   )− x )                            (7
                                                     x →+∞                         x    4
                                                                                   34
                                                               x +1
                                            f (x ) = arctan(       )          ‫ا ا‬
                                                                x
                                                                             . Df ‫د‬             (1
                                             . ]0, +∞[             f ‫ر‬         g                (2
                       .               ‫ل‬           ]0, +∞[                        g ‫أن‬          (a
                                                                             −1
                                                                    . f           (x ) ‫د‬        (b
                                                                                   35
                                                      :                ‫ ا‬IR
                  x −1
                   2
                                        π                                                  π
       Arc tan(       ) + Arc tan(x ) =      (2   Arc tan 2x + Arc tan(3x ) =                   (1
                   x2                   2                                                  4
                                                                                   36
                                                    π
(E ) : arctan(x − 1) + arctan x + arctan(x + 1) =              ‫د‬       ‫ ا‬IR
                                                     2
 . ]0,1[     ‫إ‬               ‫ا وأن ه ا ا‬      ‫و‬            (E )          ‫د‬         ‫ا‬           (1
                                                          . (E )         ‫د‬             ‫ا‬       (2

								
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