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

ASYMPTOTES.

143

du
of u and - are finite quantities A and B; then the limiting
dT
form which the equation to the tangent assumes is
y, = Am, + B,
which therefore is the equation to the asymptote.
211. It is very easy to find A and B in all cases from Howto find
1 A and B.
the equation to the curve, by putting x = -, and y =-in
that equation, and so finding a relation between u and z: then
A being the limiting value of u when z approaches zero may
be found by putting z = 0, and B, which is the limiting value
du
of - when z approaches zero, may be found by differentiating
the equation and putting z = o.
Let x3 + 3axy - y3= 0 be the equation to the curve; then Example.
putting -~ y = - and multiplying by z3 we find
1 + SaUz - 3 = 0
and differentiating Sau + (3aw - S2t) = 0;
du
in these equations put z=0, and.. u=A, =B, and we find
dx
1 - A3, SaA - AB = 0;.A. A=1, B= X, oe
and the equation to the asymptote is therefore
y = m + a.
212. If A and B turn out to be impossible or infinite, There may
then there is no asymptote corresponding to an infinite value of be ante
asymptote
the abscissa: but there may be one corresponding to a finite corresponding to a
value of x, for the point of contact may go off to an infinite finite value
distance from the origin for a finite value of x, y of course must beh
becoming infinite. Such an asymptote cannot be found by the foundw.
otherwise.