An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 9. 1o.

Continuous series of numbers.

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This second property must await closer discussion in subsequent
examples; it does not preclude the terms of the series still oscillating on from any place, that is, becoming at one time greater and
at another smaller. But when in the series of numbers only arbitrarily large values occur, yet the other property is not satisfied by
all the terms, the variable becomes indeterminately infinite.
We denote variables by the last letters of the alphabet x, y, a;
and quantities with fixed or constant values by the first letters a, b, c.
10. The principal laws for calculating with variables result from
the rules established in Chapter II for limiting values; for, the numbers
which there approximate discontinuously to the limiting value, occur
among the other values in the continuous variation. Here we repeat
the equations, written now with a reference to continuous variability,
in order to append some special theorems to them:
(I) Lim (x + y) = Lim (x) ~ Lim (y).
This equation asserts: The limiting value to which the sum of
two continuous variables tends, is equal to the sum of the limiting
values to which the summands tend, and conversely. This proposition
can be extended to several summands.
Special theorem: The sum or difference of two infinitely small
quantities is itself an infinitely small quantity, that is, one converging
to zero.
(II) Lim (x - y) = Lim (x). Lim (y).
Special theorem: The product of a finite and of an infinitely
small or the product of two infinitely small quantities is itself an infinitely small quantity, that is, one which converges to zero.
(III) Lim ()== Lim (y)
mY Lim (y)
If m denote a constant and x a variable which assumes only positive
values, then
(IV) ' Lim (xo) = (Lim x)n.
If b be a positive constant, x an arbitrary variable, then:
(V) Lim (bx) = bLim (x)
Powers with a variable exponent are called "Exponentials" in
contrast to powers with a constant exponent.
In equation (III) the condition is assumed, that Lim (y) is not
zero. But the introduction of the conception of continuous variability
furnishes an expedient which renders it possible henceforth to take
account in calculations, of quotients even with an infinitely small divisor. For, zero being no longer defined only by the
difference a - a, but as the limiting value of a series of numbers,
HIIARNACK, Calculus. 2