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 II.

THE DISTINCTION BETWEEN AN ACTUAL AND A LIMITING VALUE
EXPLAINED. A TANGENT DEFINED. THE NATURE OF THE
DIFFERENTIAL CALCULUS STATED.
The dis- 19. THE Examples brought forward in the preceding
tinction chapter, for the purpose of shewing that a function may
actual become illusory when the variable receives a particular value,
value and a
limiting lead us to make a very important though simple distinction,
namely, the distinction between an Actual Value and a Limiting Value of a function.
An Actual Value of a function f(x) is the result obtained by giving x some particular value, and performing
upon it the operations represented by f.
A Limiting Value of a function f(x) is that quantity
from which we may make f (x) differ as little as we please,
by making x approach nearer and nearer in magnitude to
some particular value without actually becoming equal to it.
That the 20. An actual and a limiting value thus defined seem
limiting
value does at first sight to be the same thing; and indeed, as long as the
noteadefiteto operations represented by f lead to a definite result, they are
quantity identical, as we shall prove presently; it is only when the
when the
actual function becomes illusory that the distinction between them is
shewn byo real, and it consists in this, that the actual value ceases to be
examples. a definite quantity, whereas the limiting value does not. This
we shall shew in the case of the examples just alluded to.
Ex. 1. 21. Let us consider the first example, namely
A fraction
which assumes the ) f ( --, (see 14).
form i
We have seen that the operations here denoted by f give no