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

108 METHOD OF FINDING - FROM
dx
Now put y = ux in (1), divide out x, and we find
A+ Bu +.x (C + Du E + E ) (FT....)...... (),
here let x = 0, and then u must become u0, if therefore u0
be a finite quantity we have
(3).... A + Buo = 0, and.. u,
B;
which gives u,.
If it should happen, however, that both A and B are zero,
0
the value we have here obtained for u0 assumes the form -,
and therefore we cannot thus arrive at the real value of u0.
But then the equation (2), dividing out x, becomes
C+ Du + E + (F + &c....)... =0;
here put x = 0, and therefore u = uo, and we find (supposing
u0 not infinite),
C D + EuDo+ E = 0;
and.~. u0 - = D -;
o = E /4 E? 4 E
thus it appears that - has two values in this case, when x =0.
dx
How the In this process we suppose u, to be a finite quantity;
values of o therefore, if u, admits of infinite values, this method does
are to be not give them, and we are left in ignorance as to whether
there are such values or not*. But we may easily determine
this in the following manner:
Put rx= Y instead of y = ux, and the equation (1) becomes (supposing A and B zero)
1 1 1
C - +D-+E+y(F...&c.), &c.=O;
u2 U Ul' '
* In the common method of finding d'/ the infinite values are sometimes overlooked.