An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

72

7LEMMAS UPON WHICH THE APPLICATION

f(a -f () f' (x)
P (a) - p (x) q (I)
ad libitumn, by sufficiently diminishing - a; hence by
f' (i)
Lemma VI, Cor. 2, the limiting value of, is also the
f f(a) f (x f f(0 _
limiting value of f(a) f() or f() f(a) when x
p (a) - (is) p (v) - (a)
approaches a. Q. E. D,
Cor. 1. 119. If a be a value of x which makes the quantities
fl (a), f (a).. f -l(a), l1 (a), 02(a)...fn'-l(a), each zero,
fP (a f"(a)
and if () be not illusory; then is the limiting
f (x) - f (a))
value of f (x) - () when x approaches a.
(x) -0(a)'
For, by the Lemma, when x approaches a,
f (x) -.f (a)_ f' (~)
lim. val. of f () -lim. val of
0<?) - c(P(a) ^(')
or of f' () -f' (
%P' (v) - ' (a)
since f' (a) and )' (a) are each zero,
f2 (0)
-lim. val. of by the Lemma,
" (c)
fo (o ). f (a)
or of,P2 (i) - ' (a)'
since f2 (a) and 02 (a) are each zero;
and so on, till we come to
f n- (v) -.f",- (a)
lim. val. of (a)
f - ( 1) -(P)(a)
which =~ (), by Lemma XIXo
^ (a)