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INFINITESIMALS In chapter 3, we found the slope of a curve at a given
point by taking very small increments of
y and
x, and
the slope was said to be equal to
. This
section will be a continuation of this concept. DEFINITION A variable that approaches 0 as a limit is called an infinitesimal. This may be written as
or
and means, as recalled from a previous section of this
chapter, that the numerical value of V becomes and remains less than any
positive number
. If the lim V = L then lim VL=0 which indicates the difference between a variable and its
limit is an infinitesimal. Conversely, if the difference
between a variable and a constant is an infinitesimal, then the variable
approaches the constant as a limit. EXAMPLE: As x becomes increasingly large, is
the term
an infinitesimal? SOLUTION: By the definition of infinitesimal, if
approaches
0 as x increases in value, then
is an infinitesimal. We see that
and is
therefore an infinitesimal. EXAMPLE: As x approaches 2, is the expression
an infinitesimal? SOLUTION: By the converse of the definition of
infinitesimal, if the difference between
and 4 approaches 0, as x approaches 2, the expression
is an infinitesimal. By direct substitution
we find an indeterminate form; therefore, we make use of our knowledge of
indeterminates and write
and
The
difference between 4 and 4 is 0, so the expression
is an infinitesimal as x approaches 2. SUMS An
infinitesimal is a variable that approaches 0 as a limit. We state that
and
, in
figure 42, are infinitesimals because they both approach 0 as shown. Theorem
1. The algebraic sum of any number of infinitesimals is an infinitesimal. In
figure 42, as
and
approach 0, notice that their sum approaches
0; by definition this sum is an infinitesimal. This approach may be used for
the sum of any number of infinitesimals.
Figure 42.Sums of infinitesimals. PRODUCTS Theorem
2. The product of any number of infinitesimals is an infinitesimal. In
figure 43, the product of two infinitesimals,
and
, is an
infinitesimal as shown. The product of any number of infinitesimals is also an
infinitesimal by the same approach as shown for two numbers. Theorem
3. The product of a constant and an infinitesimal is an infinitesimal. This
may be shown, in figure 43, by holding either
:
or
constant and noticing their product as the
variable approaches 0.
Figure 43.Products of infinitesimals.
CONCLUSIONS The term infinitesimal was used to describe the term
x as it
approaches zero. The quantity
x was
called an increment of x, where an increment was used to imply that we made a
change in x. Thus x +
x
indicates that we are holding x constant and changing x by a variable amount
which we will call Ax. A very small increment is sometimes called a differential.
A small
x is
indicated by dx. The differential of
is d
and that
of y is dy. The limit of
x as it
approaches zero is, of course, zero; but that does not mean the ratio of two
infinitesimals cannot be a real number or a real function of x. For instance,
no matter how small
x is
chosen, the ratio
will still
be equal to 1. In the section on indeterminate forms, a method for
evaluating the form
was shown.
This form results whenever the limit takes the form of one infinitesimal over
another. In every case the limit was a real number. 
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