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Limits of functions

Limit of a function. Some remarkable limits.
Infinitesimal and infinite values. Finite limit.
Infinite limit. Notion of an infinity.

Limit of a function. A number L is called a limit of a function  y = f ( x ) as  x  tends  a :


if and only if for any  > 0 one can find such a positive number  = ( ), depending on  ,  that  | x – a | < implies that  |  f ( x ) – L | < .

This definition means, that  L is a limit of a function  y = f ( x ) , if the function value approaches unrestrictedly to L , when the argument value x approaches  a. Geometrically it means, that for any  > 0 it is possible to find such a number , that if x is within an interval ( a, a + ), then a function value is within an interval ( LL + ). Note, that according to this definition, a function argument x only approaches a, not adopting this value! It must be considered at calculating limit of any function at a point of its discontinuity ( i.e., where this function doesn’t exist ).

E x a m p l e .  Find :

S o l u t i o n .  Substituting  x = 3  into the expression  we’ll receive a meaningless
                       expression ( see "About meaningless expressions" in the section "Powers and roots"
                       of the part "Algebra"). Therefore we’ll solve in a different way:

                                                        

                       Here the fraction canceling is valid, because x 3 , it only approaches 3.
                       Now we have:

                                                            

                        because, if  x  approaches  3 ,  then  x + 3  approaches  6 .

Some remarkable limits.

Infinitesimal and infinite value.  If limit of some variable is equal to 0, this variable is called an infinitesimal.

E x a m p l e .  The function   y  =    is an infinitesimal, if  x approaches  4, because 

If an absolute value of some variable increases unboundedly, then this variable is called an infinite value.


 

An infinite value has no a  finite limit, but it has so called an infinite limit; this fact is written as:


The symbol  ("infinity") doesn’t mean some number, it means only that the fraction increases unboundedly if x approaches 3. It should be noted, that the fraction can be both positive (at x > 3) and negative (at x < 3). If an infinite value can be only positive at any values of  x,  it is marked in a record. For example, at  x 0  the function  y = x–2 is an infinite value, but it is positive both at  x > 0 and at  x < 0 ; this is expressed as:

On the contrary, the function  y = – x –2 is always negative, therefore  

According to this, a result of our example can be written as:

                        then this fraction approaches 1 , i.e. its limit is 1.

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