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.

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When therefore our object is to find all the asymptotes of
a curve, we must not forget to try whether there is one corresponding to a finite value of X.
213. We may consider asymptotes somewhat differently, Asymptotes
considered
and perhaps more simply, in the following manner. somewhat
differently.
By putting - for z, and - for y in the equation to the
Z %- du
curve, we find a relation between u and a; and if u and
dz
have finite values A and B when z = 0, we may by. Taylor's
Theorem assume that
u = A + Bz + R;
where R is some quantity, such that zero is the limiting value
of - (i. e. of Rxg) when - approaches zero, i. e. when x
approaches infinity.
Therefore, restoring x and y, we may assume that
y = A + B + Rw.
Now let BP, fig. 19, represent the curve, AM= x, MP y:
also let B'P' be the line whose equation is
y'= A x + B, where y' =. MP',
then PP' =y - y' = R;
therefore, since zero is the limiting value of Rx when x
approaches co, PP' may be diminished ad libitum by sufficiently increasing m. Therefore the line B'P' continually
approaches the curve BP as we go off to an infinite distance
from the origin, but never actually meets it, though we may
make PP' as small as we please. A line thus circumstanced
is called an Asymptote, and we may find its equation just as
before, since A and B are the same quantities as before*.
214. We may determine the asymptotes of a polar curve Proposias follows: tion.
* See note, page 154.
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