01limites.continuite by RedaNachi

VIEWS: 3 PAGES: 3

									                                                                                                                                     ‫ل‬                ‫ی ت وا‬           ‫ا‬                               ‫بعت‬                                ‫ا‬
                                                                                               .
                                                                                                                                             ‫ا رس‬
                                  x0                                            g                                                        f
                                                                                                                                                                                                                                          (I
                                                                                     g ( x ) = f ( x ) , x ≠ x0
                                                                                    
                                                                                    
                                                                                     g ( x0 ) = l
                                                                                                                                             :
                                                                                                                                                                                          ∞×0          ∞              0 :                                                        (1
                                                                                                                                                              +∞ − ∞
                                                                                     .                                                           (7                                                    ∞              0
                                                                                          f ( x) − l ≤ g ( x)                                    (a
              lim f ( x ) = l
                                                                x0
                                                                                                                                                                                              :                                                                                  (2
                                                                                    lim g ( x ) = 0
                                                                                                                                                                a×∞ = ∞                           ( a ≠ 0)                   +∞ + a = +∞
               x0
                                                                                     x0


                                                                      x0                            f ( x) ≤ g ( x)                              (b                                                                          −∞ + a = −∞
                    lim g ( x ) = +∞
                                                                                              lim f ( x ) = +∞
                                                                                                                                                                ∞×∞ = ∞
                        x0
                                                                                                                                                                                                                             +∞ + ∞ = +∞                                 (a ∈ »)
                                                                                                                                                                0×∞
                                                                                               x0


                                                                       x0                            f ( x) ≤ g ( x)                             (c                                                                          −∞ − ∞ = −∞
                        lim f ( x ) = −∞
                                                                                               lim g ( x ) = −∞
                            x0

                                                                                                x0                                                                                                                           +∞ − ∞
                                                       x0                      g ( x) ≤ f ( x) ≤ h ( x)                                          (d
        lim f ( x ) = l
                                                                     lim g ( x ) = lim h ( x ) = l
                                                                                                                                                                                          ∞                                            a≠0
         x0
                                                                                                                                                                                                 a
                                                                                                                                                                                            =∞     =0                                      =∞
                                                                      x0                        x0



                             .                                                                                    (II                                                                     a      ∞                                      0
                                                                                                                                                                                          ∞    0
                                             .                                                                                       (a (1
                                              .                                                                                      (b                                                   ∞    0
                                                  :                                                        f                         (a (2
                                                                                                                                                                             :                                                                                                   (3
. f ( ]a, b]) =  lim f , f ( b ) (* f ([ a, b ]) =  f ( a ) , f ( b ) (*
                 a
                                
                                  +                                                                                                                                                                                                                                f ( x) ∞
                                                                                                                                                                                                                 .                                           lim            =    (a
                                                  :                                                           f                       (b                                                                                                                         ∞    g ( x) ∞
                                                                                                                                                                                                                                     lim ( f ( x ) + g ( x ) ) = +∞ − ∞
   . f ( ]a, b[ ) =  lim f , lim f  (*
                                                                                                                                                                                                                                                                                 (b
                     b
                                   
                                      −          +
                                                                                f    ([ a, b]) =  f ( b ) , f ( a ) (*
                                                                                                                   
                                                                                                                                                                                                                                      ∞

                                                                                                                                                                             g ( x)           f ( x)
                               a
                                                                                                                                                                                                                                                                                 (*
                                                                                                                                                 (3
                                                                                                                                                                                                                                          .
                                                          [ a, b ]                                        f                          (a                                      g ( x)           f ( x)
( ∃c ∈ [ a, b]) : f ( c ) = λ                                                                                                                                                                                                                                                    (*
                                                       f (b )              f (a)                                                     λ
                                                                                                                                                                                                                                 .
                                                                                                                                                                                                                                                        f ( x) a − a 0
                                                                                                                                                                                              .                             (a ≠ 0 )              lim          =    =            (c
                                                                [ a, b ]                                                                     (b                                                                                                         g ( x)
( ∃ c ∈ ]a , b [ ) : f ( c ) = 0
                                                                                                                  f                                                                                                                               x0             0    0
                                                                                               f ( a ) ⋅ f (b ) 〈 0                                                                                                                                     f ( x) 0 + 0 0
                                                                                                                                                                .                                                           (a ≠ 0 )              lim          =    =            (d
                                                                                                                                                                                                                                                   x0   g ( x)   0    0
                                              ]a, b[                                          f ( x) = 0
                                                                                                                                                                           
                                                                                                                                                                            x = x ;x ≥ 0
                                                                                                                                                                                   2

                            . c ∈ [ a, b]                   f ( a ) ⋅ f (b ) ≤ 0                                          (* :                                                                                                           x2 = x :                               (e
                                                                                                                                                                            x = − x2 ; x ≤ 0
                                                                                                                                                                           
                    .              c                                                      f                               (*
                                                                                                                                                                                                                                              .                                  (4
                                                                                          (III                                                                        1 − cos ( ax )              1                   tan ( ax )                                      sin ( ax )
                                                                                                                                                               lim                            =            lim                   =1                         lim                  =1
                                                                                                                                                                           (ax )
                                                                                                                                                               x →0                   2                    x →0                                             x →0
                                                                                                                                                                                                  2                      ax                                              ax

                                                                                                                  :                              (1                                                                                                              .               (5
                                                                       I                                                         f   (*                                       lim f ( x )                              x0                               f                        (a
                                                                                                                                                                                 x0
    J                   I                     f                        I                                                     f       (*                                                                .x                                 f             lim f ( x ) = f ( x0 )
                                                                                                                                                                                                            0
                                                                                                               f (I ) = J            (*                                                                                                                     x0


                                                                                                                                                                                                                                                                 f               (b
               :                           f −1 : J → I                                                               f

                                 ( ∀x ∈ J )( ∀y ∈ I ) : f −1 ( x ) = y ⇔ f ( y ) = x                                                                                                                                                                    .
                                                                           J                                          f −1            (a (2                                                                                                                                      (6
        .f                                                      J                                                 f −1               (b                                                   f                                 x0                                               f
                                                       C f −1              Cf                                     ..                 (c                             lim f ( x ) = l ∈ »                         lim f ( x )                                      x0
                                                                                                                                                                      x0                                         x0


                                                                                                      . (∆) : y = x
                                                             *
                            »           r'     r          IR +              b       a            (j
                                                                                                                                                                                                    .       f −1                                                    (3
                 (a )                                       a r ⋅ a r ′ = a r +r ′
                        r′
                    r
                             = a r r′                                                                 I                                                                                                                                              f
                                                                            r                                           J = f (I )                                                              f −1                        ( ∀x ∈ I ) : f ′ ( x ) ≠ 0
                                                                         a
          (a b )
                    r
                        = ar ⋅b r                                            = a r −r ′
                                                                         ar′                                                       ( ∀x ∈ J ) : ( f −1 )′ ( x ) =
                                                                                                                                                                                                    1
                        r
                                                                                                                                                                                            f ′ ( f −1 ( x ) )
                 a a
                        r
                                                                            1
                   = r                                                       = a −r
                 b  b                                                     ar                                              (n ∈ » )       *
                                                                                                                                                      n                                                                                      (IV
    Arc tangente                                                                (V                    IR +              y                                         n
                                                                                                                                                                      x                     »+                          x                    :                      (1
y                   Arc tan ( x )                     »          x              :                (1                                                                                                         .y =x           n


                                         . tan ( y ) = x                             π π                                                                . 2 ≥ 0 ‫61 = 2 و‬                                                                                   (* :
                                                                                                                                                                                       4                                        4
                                                                                                                                                                                                                                    16 = 2
                                                                                    − 2 , 2 
                                                                                            
                                                                                                                                                +
                                                                                                                                                                          16 ≠ −2                               ( −2 )                      = 16
                                                                                                                                                                                                                                        4
                                                                                                                            −2 ∉ »                                    4
                                                                                                                                                                                                                                                             (*
                                                                                :                (2
                                                                                                                                                                                                                                                                    (2
                                             »                            Arc tan                (a
                                                                                                                  ( ∀x ∈ » ) : x ≥ 0 (b
                                                                                                                              +        n
                                                                                                                                                                                            »       +                                                n
                                                                                                                                                                                                                                                                    (a
                                                                 π                          π
                                               ( ∀∈ » ) : −    〈 Arc tan x〈 (b                                            .
                                                                                                                            ( ∀x , y ∈ » ) :                      +
                                                                                                                                                                          *)
                                                                                                                                                                                   n
                                                                                                                                                                                       x = y ⇔x =y
                                                                                                                                                                                                n
                                                                                                                                                                                                                                                                    (c
                                                             2              2
                                                                                                                                                                          *)
                                                                                                                                                                                n
                                                                                                                                                                                       x <n y ⇔x <y
                                             ( ∀x ∈ » ) tan ( Arc tan x ) = x (c
                                                                                                                                       .(
                                                                                                                                               ∀x , y ∈ » + ) :                *) x = y ⇔ x = y
                                                                                                                                                                                   n   n

                                                                                                                                                                                                                                                                    (d
                                       π π                                                                                                                                 *) x 〈 y ⇔ x 〈 y
                                                                                                                                                                                   n   n

                                 ∀x ∈  − ,   : Arc tan ( tan(x ) ) = x (d
                                       2 2                                                                  ( ∀x , y ∈ » ) :                 *) x = y ⇔ x = y
                                                                                                                                                      n   n

                                                                                                                                                                                                        :                                        n                  (e
                 ( ∀x , y ∈ » ) :        *) Arc tan x = Arc tan y ⇔ x = y                                                                         *) x 〈 y ⇔ x 〈 y
                                                                                                                                                        n n

             .                                                                                   (e
                                         *) Arc tan x 〈 Arc tan y ⇔ x 〈 y
                            π π 
                                                                                                              ( ∀x , y ∈ » ) :                 *) x = y ⇔ x = y
                                                                                                                                                   n   n

                                                                                                                                                                                                    :                                                               (f
                  ∀x , y ∈  − ,   :               *) tan x = tan y ⇔ x = y
                                                                                                                                                                                                                                                     n
                                                                                                                                               *) x 〈 y ⇔ x 〈 y
                                                                                                                                                   n   n
                            2 2                                                              (f
                                                      *) tan x 〈 tan y ⇔ x 〈 y                                                                                        ( ∀x ≥ 0 ) . n x n                                ( x)
                                                                                                                                                                                                                                             n
                                                                                                                                                                                                            =               n
                                                                                                                                                                                                                                                 = x (* (g
                                               :                          arctan                 (g
                 ( ∀x ∈ » ) : Arc tan ( − x ) = − Arc tan ( x )                                                             (∀x ∈ IR ): n x n = x                                                                                       n                      (*
                                                                                                                                   +                                                                *
                                                                                                 (h                          »                      b                 a                    IN                           p                    n                      (h
                                                                                                                                                                                                            n
                                                                                                                                                                                                                a . b = ab  n                    n
                                                                                                                                                                                                                                                               (*
                                                                                                                                                                                                            ( a)
                                                 1                                                                                                                                                                                  p
                                                                                                                                   ap = n a                                                                                                 = n ap
                                                                                                                              np
         x                      0                 3
                                                                 1                      3                                                                                      ;                                n
                                                                                                                                                                                                                                                               (*
                                                π                π                   π
    Arc tan ( x )
                                                                                                                                                                                                                            n
                                                                                                                                                                                                                                    a                    a
                                0                6                   4                3
                                                                                                                            n p
                                                                                                                                   a =
                                                                                                                                                 np
                                                                                                                                                      a                    ;                            ( b〉 0 ) n                          =        n     (*
                                                                                                                                                                                                                                    b                    b
                                                                                                                                                                                                            a. a =                               an+ p
                                                                                                                                                                                                                                            np
                                                                                                                                                                                                        n           p
                                                                                                                                                                                                                                                               (*
                                                           a =b                                                                                                                                                             p
                                                                                                (a
                                                                                                                ( n ∈ » , p ∈ » ) ( ∀x
                                                                                                                              *
                                                                                                                                                                                       〉 0) : x                             n
                                                                                                                                                                                                                                        = n x p (* (i
                                        π π
                                a, b ∈  − ,                            tan ( a ) = tan ( b )                                                                                              p
                                        2 2                                                                                . ( ∀x ∈ » ) : x = x n :
                                                                                                                                           n p
                                                                                                                                                                                                                                        p                      (*
                                .
                                         arctan(a ) = b                                     (b                                                                                                                                                           :
                                   π π
                              b ∈ − ,                                   tan (b ) = a                        x
                                                                                                                =
                                                                                                                    n   x
                                                                                                                                                              x y = n x ⋅n y                                                xy 〉 0                                  (1
                                   2 2                                                                  n                                             n
                                                                                                              y     n   y
                                                                                                           x = 3 x 3 = 3 x 3
                                                                                                                              ( )                                     ; x ≥0
                                                                                                          .
                                                                                                                                                                                               ( ∀x        ≥ 0) : 3 x 3 = x                                  (* (2
                                                                                                                                (                         )
                                                                                                                                                              3
                                                                                                           x = − 3 −x 3 = − 3 −x                                     ; x ≤0
                                                                                                           


                                                                                                                                    a 3 + b3                                                                  a 3 − b3
                                                                                                                    a+b =                                                                  a −b =                                                              (*
                                                                                                                                  a − ab + b 2
                                                                                                                                       2
                                                                                                                                                                                                            a + ab + b 2
                                                                                                                                                                                                                2


                                                                                                                                                                                      a 4 − b4
                                                                                                                                                                  a −b =
                                                                                                                                                                               a 3 + a 2b + ab 2 + b3

								
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