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 IV.

CERTAIN LIMITING VALUES DETERMINED WHICH WE SHALL
REQUIRE TO KNOW HEREAFTER.
LemmaIX. 44. IF s be the length of any arc of a curve, and c
S
the length of its chord; the limiting value of - when s
approaches zero is unity.
Let PSQ be the arc (fig. 7), PCQ the chord, draw PR
the tangent at P, and QR perpendicular to PR, and let
z RPQ = w: then, as long as P does not actually coincide
with Q, we may evidently assume that PSQ lies between
PCQ and PR + QR (supposing of course that Q is taken
near enough to P, so that the curve shall always bend towards the same side between P and Q); i.e. we may assume
that
s lies between c and c cos w+ c sin w;
and therefore that - lies between I and cos w + sin w.
c
Now, by the definition of a tangent in (25), 0 is the
limiting value of w, and therefore 1 that of cos + sin w, when
s approaches 0: therefore by Lemma VII, 1 is the limiting
value of - when s approaches 0. q. E. D.
c
Lemma Xo 45. If 0 be an angle measured by the subtending are
0 0
of a circle, the limiting value of i —, and that of --
when 0 approahsin tan i0
when 0 approaches rero, are each unify.