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.

~ 20.

Limiting value of the quotient of differences.

37

function at this point can be imagined as a curve, i.e. a geometrical figure having a definite direction at the point, so that its
course at this point is represented as approximately as we can wish
by a determinate right line. Therefore in geometrical language the
condition that there should be a limiting value, is no other than the
condition that a function should admit of being represented by a
curve. (~ 15.)
On the other hand the limiting value becomes infinitely great in
a determinate manner, when the measure of the deviation from the
right line PP1 increases always positively or always negatively beyond
any finite amount or tan a' becomes infinite (~ 24a). But it becomes
completely indeterminate when the deviation is neither zero nor becomes infinite in a determinate manner, but oscillates between different
limits.
This limiting value, defined by the equation:
dq -. f(x$l O Q )-/f(x). *f f (x ) sx)-f(x)
(IIP) dy Lir f(-x+Ax) —f( or more briefly: Liam fx -f(x
- dx 00 OAx Jx0o A
called the Progressive Differential Quotient of the function at the point x,
affords a measure for the change of the function at the point, when x
increases.
Likewise the Regressive Differential Quotient:
(IIa) d -1 Lim f(x Ax) -f(x) or more briefly: Lim f(x- )-f()
d /X 0-0 OAx Jx-0 -AX
affords a measure for the change of the function at that point when x
decreases.
Instead of equations (III), provided dY is finite, we can also
write Ay - dy,
Ax dx,
where 6 is precisely that difference between the terms of the series
formed of the quotients of differences and the limiting value of that
series which converges to zero with Ax. Here it is to be remarked,
that since the quotient of differences is a continuous function of Ax,
a finite value of Ax can always be found which shall satisfy this
equation, however small but finite be the value of (; but for a given
6 this has not to be the same Ax for all values of x in an interval
(see next paragraph). The equation
Ay dy
Ay —AX.- + Ax. 5
then asserts: the smaller we choose Ax, the more accurately is the
corresponding change of the function equal to the product of the
differential quotient by Ax, so that for the point itself this precisely
denotes the measure of the increase. It is also evident, as stated at