Limites al infinito:    propiedades

·        Lim       k        =     0      X →∞  

·         Lim                k      =   0       X → + ∞        

 

·        Lim            k        =     0      X → - ∞   

 

·        Lim            1        =     0     X → ∞       x

                                                                                            -    2x   +   5

·         Lim               =    lim                                     

      X → +∞       8              x → +∞             8

                                                                                                       

                                                        = lim                     1  -  2     + 5

                                                           x →+∞               x  

                                                                                      8  +  1  +  2

                                                                                                 

                                                                                     X      -     9

·        Lim                   x -  9                             =               x            x     .   

      X →∞                                     

                                                                                            

                                                                  

                                                                                                     9

                                                              =   lim                 1   -     x         .

                                                                   X → ∞          + 3   +   2    

                                                                                                   X      

                                                              =  lim               1    .

                                                                 X →∞       

                                                              =       1

                                                                       2

                                                                                      (

·        Lim                              =       lim               x        .

      X → ∞         x                                       x → ∞          x

                                                                                         x

                                            

                                                                                        (

                                                                =  lim                 x  .

                                                                    X→∞             1

                                                                =            0

                                                                              1

                                                                =         0

LN  = logaritmo natural     =    ln ( 10 )

1.)           Lim          ( ln x – x + cos x)

             X→10

        =    lim       ln x  –   lim x      +     lim cos x

              X→10               x→10           x →10

        =  - 8.53

2.)      Lim              1       =   0.11920

        X→2       1 +  

3.)      Lim

        X →2     ( 2    + 3      

  

                                                   = 26,77

4.)    Lim

      X →π    (  cos 3  x   +  1  sen  2 x +   tan  x )  = 0.00000

                              2           4

5.)     Lim                                     

       X →3               

                                                   =  

                                                            10

                                                   =    126 

DERIVE

Y  = 3   - 5

Y =  0 3.2  - 0

Y` = 6x  =  f (x)  =  6x  =  dx  =  6x 

                                         dx

y =  k

y =  k

y = k        y = f (x)

y`=  0

y`= f`(x)= dy

                 dx

f(x) = 1   - -   - x + π

          2

f`(x) =    1 . 4   - 1 . 3   - 1 . 2   -  1 . 1  

              2

F`(x) = 2  - 3   -2x -1x

F`(x) = 2   - 2x -1   

Y = 2   +   x  +  e

                          2

Y = 2  + 1 x + e

                  2

Y`= 2 .  1  + 1 . 1  +0

             2              2

Y`=     +   1

                       2

Interpretacion geometrica de la derivada

M =  Δy = yf – yi = y1 – y 2 = f (x1) f ( x2)

      Δx    xf  - xi = x1 – x 2 =   x1   -  x 2

La derivada de una function es la pendiente (inclinacion) de la recta tangent a una curva en u punto.

Lim  m sec =   lim          f ( x + h ) + f ( x )

H→0          h→0             h   

                  =  f`(x)        ly =   m tan   =  Y` 

                                     lx

hallar la pendiente de la recta tangente (derivada) de la función: luego hallar la recta tangente en el punto  ( 3 , 4 )

f(x) =

                       = lim         [( x + h  [

                          H →0                                                 h

                       = lim        

                          H →0                                         h

                       =lim             2xh +  - 2h

                         H→0                   h

                       =lim              h ( 2x + h – 2 )

                         H→0                      h

                       =lim       ( 2x + h – 2 )

                         H →0