3.4.3. Infinitesimals and bounded functions.
Definition.
A
function f(x)
is
said to be infinitesimals as
х
а
if, for any М,
there exists a
such
that
whenever
.
Notation:
.
Definitions. 1. A function f(x) is said to be bounded on a domain D if, for any х from D,
|f(x)|M.
For example, f(x)=cosx , |cosx|1, M=1,
.
If this condition is not satisfied, then the function is said to be unbounded, i.e., this function is an infinite quantity.
2. A function f(x) is said to be bounded as х а if, for any х from a neighborhood of а, |f(x)|M.
3. A function f(x) is said to be bounded as х if, for any х>N, |f(x)|M.
Theorem I. If a function f(x) has a finite limit as xa then f(x) is bounded as x a.
Proof. It follows from the definition of the limit of a function that, whenever |x–a|<, we have |f(x)–b|<, or b–<f(x)<b+. Therefore, |f(x)|<|b+|=M. Thus, if |x–a|<, then |f(x)|<M, i.e., the function is bounded by definition.
Theorem
II.
If a function has a limit as
х
а
and this limit is not equal to zero, then
is
bounded as
х
а.
Proof.
Suppose that
.
It
follows from the definition of the limit of a function that if
, then |f(x)-b|<,
i.e.,
b–<|f(x)|<b+.
Hence
.
Thus,
for any x(a-,a+),
we
have
,
i.e., the function
is
bounded.
3.4.4. The infinitesimals and their properties.
Definition 1. A function (х) is called an infinitesimal as х а if
.
Definition 2. A function (х) is called an infinitesimal as х а if, for all х( а-; а+),
.
These two definitions are equivalent, i.e., we can obtain the second definition from the first and vice versa. (Prove this).
Theorem I. If a function f(x) is represented as the sum of a constant number and an infinitesimal, i.e.,
f(x)=b+(x), (2)
then it has a limit:
.
Conversely, if a function f(x) has limit b, then the function can be represented in the form (2).
Proof.
Suppose that f(x)=b+(x)
for |x-a|<,
where
.
Consider
the difference
,
where
|x-|<.
For
all
|х-a|<,
we
have
;
thus,
by
definition,
.
2.
Suppose
that the limit (1) exists then, by definition, we have
whenever
|x-a|<.
Let
.
Then,
for |x-a|<,
we
have
,
i.e., (х)
is
an infinitesimal and, by definition, f(x)=b+
(x). This
completes the proof of the theorem.
Theorem
II.
If
(х)
is an infinitesimal
as
х
а,
then
is
an infinite quantity.
Proof.
For
|x–a|<,
we have
,
where
М
is
given large number. Consider
.
We
have
=
.
Thus,
as
х
а
,
i.e.,
is
unbounded, as required.
Theorem III. The sum of finitely many infinitesimals is an infinitesimal:
1(х) + 2(х) + 3(х) + … + к(х)= (х).
Proof.
We consider infinitesimals as х
а;
since
1(х)
is
an infinitesimal, it follows that, for
|x–a|<1
, we
have
(к
is
a number).
Since 2(х)
is
an infinitesimal, it follows that, for |x–a|<2
, we
have
,
etc.
Since к(х)
is
an infinitesimal, it follows that,
|x–a|<к
, we
have
.
We
take the smallest number among 1,
2,
3,
…, к,
i.e.,
=min
{1;
2;
3;
…; к}.
Then,
for all
|x–a|<,
each
of values
is smaller than 
.
Let us estimate the infinitesimals:
hence, the sum (х) is infinitesimal.
Question-Remark. The weight of one snowflake is an infinitesimal(?). Only finitely many snowflakes (they can be counted) fall on a tree branch during a certain period of time. By the above theorem, their weight is also an infinitesimal(??). The question is, why snow breaks tree branches sometimes (???).
Theorem IV. The product (x).z(x) of an infinitesimal (х) by a bounded function z(x) as x a is an infinitesimal.
Proof.
For
|x–a|<
, we have
;
for
|x–a|<,
we
have
|z(x)|<M.
The
product of functions is estimated as
for
all
|x–a|<,
which
implies
|(x)z(x)|<1;
i.e., the product
(x).z(x)
is an
infinitesimal.
Corollary. The product of infinitesimals is an even smaller quantity.
Theorem V. An infinitesimal divided by a function having nonzero limit as х а is infinitesimal, i.e., if
,
then
is
an infinitesimal.
Proof.
Since
,
the
function
,
is bounded by Theorem II. Consider the ratio
,
which is the product of an infinitesimal by a bounded function. The
preceding theorem implies that the quotient is infinitesimal.