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

OF THIE DIFFERENTIAL CALCULUS DEPENDS.

'73

_____CC f (x) - f(a)
Hence f (a) is the limiting value of
lp" (a) cP (x) P (a)
when x approaches a. Q. ri. D.
120. If a be a value of x which makes f'.(a), f2 (a) Cor.2.
f" (a)
f n' (a) each zero, then is the limiting value of
Fr
f (x) f (a)
(x) - fa) when x approaches a.
(X - a)"
This is easily proved by putting op (x) (x - a)" in the
last Cor., and therefore c/ (x) - (/ (a) = (x - a)", /5' (a) = 0,
(P'(a)= 0... 0"-'(a) = 0, /5n"(a) = Fn; which evidently gives
fC"(aC) f( ) -f(a)
us F for the limiting value of fx)f( when x
rn2 (IV - a)
approaches a. Q. D.
Conversely: If a finite quantity A be the limiting value Cor. 3.
f (X) f (a)
of f~x) - f~a) when x approaches a, then must f (a)= 0,
Of (x-a)"
fP(a)TO f (a)-=...f"-' (a)=0, and f"(a)= Fn.Ai,
For if we have f'(a) = 0, f2(a) = 0...f''(a) = 0, but fr"(a)
not 0, r being,, any integer less than n; then the limiting
value of (x) - f(a) w (a),vla -f hen x approaches a will be
(x~ a) r rr
and therefore that of fwi -hf(a)
- am)" whica)h
f f)f(a)
will be I.f'( (Lemma VIII.); which is infinite, con0 Fr
trary to hypothesis. Hence the truth of the Cor. is evident.
121. If f' (a), f2(a)... f"n' (a), be each zero, and Cor. 4.
f" (a) not zero, then, for. all values of x taken su~ftciently
near a, f (x) f (a) has the same sign as f" (a) (x - a)