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.

108

Implicit functions.

Bk. I. ch. X.

which y is infinite, but x finite, so that the ratio y: x is infinite;
clx
here we have - = 0.
62. We conclude these considerations with the problem: Let
(p(x) and (x) be any two functions, which both become infinite for
x = a; it is required to find the limiting value of the quotient
(X) for x = a.
9p(x)
Putting x == a + -i consider 9 and 4' as functions in z, then they
become infinite for - = o, i. e. when S increases positively or negatively beyond all limits. Therefore the problem is reduced to this
other, to calculate Lir (z) for = oo or - oo, when Lim cp (z) oo,
Lim P(z) -= oo. It was shown in ~ 24d, that provided h(z ) /(z)
h
has any determinate limiting value for 2 + oo, we have then
Lim f(z) _ f(z + h) -- f()
z 71
for every value of h. If now the function f be continuous from a
point S = z,, and infinite only when == 0, if further its differential
quotient be everywhere determinate, which we know cannot be the
case unless from a certain point the function only increases or only
decreases, and if the progressive differential quotient be identical with
the regressive, then it follows that
Lim f(z) Lim f(z +h) - f(z) == Lim f'(z + 0 h)
When these formulas apply to gp and 4,, we have
Lim (z) Lim '(z + - b), Lim "() Lirm i'(zs+ O'h),
and so by division:
Li -Lim =Lim / +- Oh) for = 0, (~) > "i\z - (+ 0 /h)
i. e. Provided the derived functions p' anzd ai' have each a determinate value
for =- co the same when taken progressively as regressively, then their
quotient also has; and the limiting value of the quotient of the /-ilctions
cp anId 4, which become delerminately infinite, is equal to the quoticnt
of the derived functions.*)
Examples:
(Va +, 1x) }
1) {ab) 1}
X=CO
*) The hypotheses of the theorem can be further generalised, see
Rouquet: N. Annal. de mathem., 2. Ser., T. XVI.
Stolz: Math. Annal., Vol. XV.