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

CHAPTER X.

DETERMINATION OF THE LIMITING VALUES OF VANISHING
FRACTIONS.
148. TirE Differential Calculus may often be employed Vanishing
with great advantage to determine the limiting values of
vanishing fractions; i.e. functions which assume the illusory
0
form - when the variable receives some particular value.
0
If f(a) =0 and p (a) =0, to determine the limiting value To deter>~f/\~~~~~ (X)s~~~~ ~mine the
f(x) limiting
of ) when x approaches a. value of a
pa(x) vanishing
fraction.
Since f (a) and p (a) are zero, f) = () f()
q5 (x) -) (j) - (p (a)
<jff(<a?) f f (a)
hence by Lemma XIX. the limiting value of ( ) is f((a)
P(e) ^(a)
or more generally, by Lemma XX. the limiting value of
-f(I) when x approaches a is also the limiting value of
f ( IV) _)
f (X); if it should so happen that f,(a) is not an illusory
^ (x) r (p(a)
quantity, then it is the limiting value we seek: but if
f'(a)= 0 and 0'(a) = 0, then by the same Lemma, Cor. 1,
f2(a) f(v)
the limiting value of f ) is that of f(-, and therefore
0 2x) (;-)
f'5(a), if not illusory, is the value we seek: if however f2(a)
and 2S(a) be both zero, then we must try the third differential coefficients, and so on till we come to two differential
coefficients which do not both vanish (nor of course become
both infinite) when x = a; and thus we shall obtain the
limiting value we seek.