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Sec. 4] Infinitely Small and Large Quantities 33
Sec. 4. Infinitely Small and Large Quantities
1. Infinitely small quantities (infinitesimals). If
i.e., if |a(x)|a
< fi(e), then the function a (x) is an
infinitesimal as x a. In similar fashion we define the infinitesimal a (x)
as x * oo.
The sum and product of a limited number of infinitesimals as x +a are
also infinitesimals as x-+a.
If a(x) and p (x) are infinitesimals as x *a and
lim SlJfUc,
x-+a P (x)
where C is some number different from zero, then the functions a(x) and p(x)
are called infinitesimals of the same order; but if C = 0, then we say that the
function a (x) is an infinitesimal of Higher order than p (x). The function
u (x) is called an infinitesimal of order n compared with the function p (x) if
where < J
If
C| < -f oo.
lim
Q(x)
" -C '
then the functions a (x) and p (A*) are called equivalent functions as x *a:
For example, for x > we have
sinx~x; tanx~ x; ln(l-fx)~ x
and so forth.
The sum of
term whose order
two infinitesimals
is lower.
of different orders is equivalent to the
The limit of a ratio of two infinitesimals remains unchanged if the terms
of the ratio are replaced by equivalent quantities. By virtue of this theorem,
when taking the limit of a fraction
lim
!> , aPW
> where a (x) >.0 and p (x)
the numerator or denominator
the resultant quantities
as x *a t we can subtract from (or add to)
infinitesimals of higher orders chosen so that
should be equivalent to the original quantities.
Example 1.
,. j/?T2? -a/7
,.
'
lim i- = lim
*-*o 2x 2
ber
2.
Af
Infinitely large
there exists a
quantities (infinites). If for an arbitrarily large num-
6(N) such that when < x a | | < 6(N) we have the
inequality
lfMI>tf.
then the function f(x) is called an infinite as x >a.
2-1900