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

VANISHING FRACTIONS.:101

f(x) tan - sin wx a o 0 Example 2.
Let =, which assumes the form -
)p (V.) M, 0
when x = 0,
f' (x) sec2 - cos x 0
-= -() =- when = 0
(p (x') 3 x2
1 1 - COSa3?
cos: ~ 3 x2
150. Here we must make a remark of some importance. Important
simplification we may
f' ( ) often make'
Suppose that we obtain ( in the form UV, and that in the proq (V,) cess.
we know A to be the limiting or actual value of U corresponding to the value of x which makes the fraction vanish;
then, by Lemma VIII., the limiting value we require is
A x limiting value of V. Hence if in the process of obtaining
f' (a)
the limiting value of a vanishing fraction, we find that,
(' ()
has any factor whose limiting or actual value we know, we
may always substitute immediately for that factor its value
whatever it be, and so simplify the operation.
Thus in the present example the factor = 1 when
COS X
= 0, we may therefore substitute 1 for it, and then we have
1 _ COS3 V
only to find the limiting value of - cos; i.e. of
d (l - cos3 cos Cos2 o sin x
or
d (3 x") 2
and here again putting 1 instead of the factor cos2,v we have
sin x d sin x
only to find the limiting value of; i.e. of
2ox d2x
COS X
or, which is -. Hence the limiting value required
is.
2'