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

18

LEMMAS RESPECTING LIMITING VALUES.

We have supposed that the function f(x) has only one
value for each value of Ix; if however it has more than one,
n values suppose, it is evident that there will be n different
limiting values, and no more, when x approaches a. Thus if
f( )=a+b e f _ )
which has two values for each value of x, then there are two
limiting values when v approaches 1; viz. a + b and a - b.
LemmaVI. 39. If f(x) and ( (x) be two functions, one of which,
f (x), becomes illusory when x = a certain value a, and the
other, p (x), does not; and if we can shew that f (x) = ( (x)
for all values of x, a of course excepted; then p (a) is the
limiting value of f(x) when x approaches a.
For by (29), we diminish (p (Z) - (a) ad libitum by
sufficiently diminishing x - a; but in so doing we never
suppose x to become actually equal to a, and we are therefore
sure that f(v) = p (Ix); therefore, we diminish f(x) - 4 (a)
ad libitum, by sufficiently diminishing x - a; i.e. (p (a) is
the limiting value of f(x) when x approaches a. Q.E.D.
Cor. 1. If ( (a) be illusory as well as f(a), and if we know A to
be the limiting value of i (x) when.r approaches a; then we
may shew, in exactly the same way, that A is also the limiting
value of f(x) when x approaches a.
Cor. 2. If, instead of being able to shew that f(v) = (p (x) for all
values of x except a, we can prove that f(x) - (x) is diminished ad libitum by sufficiently diminishing x - a; then
the same conclusions evidently follow; that is to say; the
limiting value of f(v) when x approaches a is (p (a), or A
if tp (a) be illusory,
Lemma 40. If f (x) be a function which becomes illusory when
x = a, and if we can prove that f (x) lies between another
function (p (x) and a constant A for all values of x taken
sufficiently near a; and moreover, that A is the limiting
* When we say that f(x) lies between p(S(a) and A, we mean that f(z) is not
greater than one of these quantities and not less than the other.