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.

18 Conception of a variable quantity. Bk. I. ch. III.
the quotient Lim () can also still retain a determinate value when, for
Lim (y)
arbitrarily small values of y, the quotient x: y expresses a number
approximating to a determinate limiting value, finite or infinite. But
this is only possible when the law of formation of the series x:y
is given, i. e. when to each number within the series x the number
in the series y is known, by which the former is to be divided. When
this is so, by the quotient Lim (x) we shall understand the value
'I~ ~ ~ Lim (y)
Lim (xy)*). If x have a determinate finite limiting value, the series
y
x: y will consist of numbers, which increase in absolute amount beyond any limit. But if the limiting value of x itself be zero, then
the series x: y can have either a finite, or an infinitely small, or even
an infinitely great or yet no definite limiting value at all, and this
is to be decided only in each individual case by forming the series.
For equations (I), (II) the law holds also, that in order that
the left side may be formed without ambiguity, the connexion between
the places of the series x and those of the series y must be given.
In this case the left sides can then still express determinate limiting
values when the limits of one or of both series for x and y transcend
any finite amount."*)
*) A simple example may make this clear to the beginner. Suppose the series
y given, in which y is to travel through all numbers from 1 to 0; let the series
x consist of the numbers 3y - y2 so that therefore for y = 1 x 2, for y =
Lir x
x -, for y= - x-a=, y - Ox =0. If now by - we understand
Lim y
h alu Liax — y_the value Lim, it will be expressed for every value of y by - — =3-y,
and thus although y, and with it x, have the limiting value 0, the series of the
quotients has the limiting value 3.
**) The last half of this chapter will become more intelligible when the investigations which immediately follow shall have furnished us with materials for
definite examples