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 VIII.
CERTAIN LEMMAS UPON WHICH THE APPLICATION OF THE
DIFFERENTIAL CALCULUS IN MANY CASES DEPENDS.
112. HAVING now concluded what may be considered
the first part of the differential calculus, namely that in which
we determine a set of rules whereby differentiation may be
performed in all cases with facility; we proceed to the second
part, in which we shall shew some of the principal uses which
may be made of the differential calculus in pure mathematics.
But we must previously prove the following lemmas upon
which the application of the differential calculus in a great
measure depends.
113. If f(x) changes its sign when x passes* through Lemma
the value a, then f(a) must be 0 or o XVII.
For f (a) is the limiting value of f (x) when x approaches
a (Lemma III), and therefore by Lemma IV f(x) has
the same sign as f(a) for all values of x taken sufficiently
near a; hence, if f(a) be positive, f(x) is positive for all
values of x taken sufficiently near a, and therefore cannot
change its sign when x passes through the value a; and the
same is true if f(a) be negative. If therefore f(x) changes
its sign when x passes through the value a, f(a) can neither
be positive nor negative; i.e. it must be o or -. E. D.
114. It is important to observe, that what we havef(x) does
proved here is, not that f () must change its sign when f(a) not necesis 0 or oo, but that if it does change its sign, f(a) must change its
sign when
be 0 or co. f(a) is0o
00.
a By saying that, "f(x) changes its sign when x passes through the value a,"
we mean that when we give x any value a little greater than a, f(x) has a different
sign to what it has when we give x any value a little less than a.