# Limits of functions

*Limit of a function. Some remarkable limits.*

Infinitesimal and infinite values. Finite limit.

Infinite limit. Notion of an infinity.

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 (
*
L
*
–
,
*
L
*
+
). 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.