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

CERTAIN LIMITING VALUES DETERMINED.

27

Hence if we consider only the first seven decimal places,
2.7182818 is the limiting value of 1 +-), when n approaches o.
50. In the same manner we might obtain this limiting Base of
Hyperbolic
value to any greater number of decimal places. It is, how- Logever, an incommensurable number like 7r, and therefore, no arith'ns;
matter how far we go, we shall never be able to obtain it
exactly in numbers. It is usually denoted by the letter e,
and it is taken as the base of a system of logarithms commonly called Hyperbolic.
For a certain reason, which we shall presently explain,
logarithms calculated to base e are the most convenient to
use in analytical calculations. We shall therefore always
suppose that e is the base of whatever logarithms we have
occasion to employ unless the contrary be specified,
51. Hence e is the limiting value of (1 + x)x when Cor. 1.
x approaches zero, x being any continuous variable.
For this limiting value by Lemma I. is the same thing
as the limiting value of (1 + z)z when z approaches 0, where
=- and n is always an integer; and this latter limiting
n
value by Lemma II. is the same thing as that of (1 + —
when n approaches co, which = e. Q. E. D.
1 *
52. Hence, logae or g is the limiting value of Cor. 2,
loga
lo0g (1 + x)
a ( + — x) when x approaches zero.
x
logr ( ) + o) o
For ga (- =log,(l + )V, and by what has just
been proved, and by Lemma VIII. the limiting value of this
latter quantity when, approaches zero is logae.

* If log, e = c, then a = e, and,'. log =nl, or C-.
I = Tog~~~~ aof