Limits and the Infinity by gqz18849

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									Limits and the Infinity


        Limits at the Infinity
        How to Compute Limits at the Infinity
        The infinity as a Limit


Index                                     FAQ
Limit at the Infinity (1)
The definition of limits has to be modified for the case where the variable
approaches  or - .

 Definition 1   A function f has a finite limit L as x if the values of the
                function f get arbitrarily close to the number L as x
                grows.

 Notation        lim f  x   L
                 x 

The same definition in a rigorous way.

 Definition 2    lim f  x   L    0 : M such that
                 x 

                                x  M  f  x   L  .



     Index                   Mika Seppälä: Limits and the Infinity
                                                                      FAQ
Limit at the Infinity (2)
 Definition 2        lim f  x   L    0 : M such that
                     x 

                                            x  M  f  x   L  .

                            sin  x 
 Example             lim                  0.
                     x       x
 Proof                Let   0 be given.


         sin  x             sin  x         1           1
                     0                          if x   M.
            x                      x           x           



    Index                              Mika Seppälä: Limits and the Infinity
                                                                               FAQ
Limit at the Infinity (3)

Definition 1    A function f has a finite limit L as x- if the values of
                the function f get arbitrarily close to the number L as x
                gets smaller.


 Notation        lim f  x   L
                 x 


The same definition in a rigorous way.


 Definition 2    lim f  x   L    0 : M such that
                 x 

                                x  M  f  x   L  .



    Index                    Mika Seppälä: Limits and the Infinity
                                                                     FAQ
How to Compute Limits at the Infinity
                      2x 3  5 x 2  x  1
 Example         lim
                 x        x3  1
To compute limits of the above type, the main method is to rewrite the
expression so that, after substitution x = , the expression can be simplified
to a number using the rule c /  = 0, where c is any finite number.
In the above example, direct substitution does not work because it leads to the
undefined expression  / .                  5 1      1
                                          2  2  3        After substitution
                     2x  5 x  x  1
                        3     2
                                             x x     x .
 Solution    Rewrite                                       x=, all the circled
                           x3  1                1
                                              1 3          terms become 0.
                                                 x
                                                             5 1    1
                                                       2       2 3
            Substitute x   to conclude                     x x   x  2.
                                                                      x 
                                                                 1
                                                              1 3
                                                                x

    Index                   Mika Seppälä: Limits and the Infinity
                                                                          FAQ
The Infinity as a Limit (1)
Definition 1    Let x0 a finite number. A function f has the infinity as its
                limit as xx0 if the values of the function f grow arbitrarily
                large as x gets close to x0.


 Notation       lim f  x   
                x  x0


The same definition in a rigorous way.


 Definition 2     lim f  x     M :   0 such that
                 x  x0

                          0  x  x0    f  x   M.



    Index                     Mika Seppälä: Limits and the Infinity
                                                                      FAQ
The Infinity as a Limit (2)
 Definition 2   lim f  x     M :   0 such that
                x  x0

                         0  x  x0    f  x   M.


                       1
 Example          lim    2
                           
                  x 0 x



 Proof          Let M  0 be given.


         1            1             1
           2
             M  x 
                   2
                         x  0x    
         x            M             M


    Index                    Mika Seppälä: Limits and the Infinity
                                                                     FAQ
The Infinity as a Limit (3)
 Problem        Following the outline of the previous definitions define:
                                        lim f  x   .
                                        x 




 Solution and        lim f  x     M : m such that
                     x 
 Definition
                            x  m  f  x   M.

 Problem        Following the outline of the previous definitions define:
                                         lim f  x   .
                                        x 


 Solution and        lim f  x     M : m such that
                    x 
 Definition
                            x  m  f  x   M.

    Index                    Mika Seppälä: Limits and the Infinity
                                                                     FAQ

								
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