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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