An introduction to mathematics, by A. N. Whitehead.

DIFFERENTIAL CALCULUS 229
that, as long as we retain anything like "h
tending to a," as a fundamental idea, we are
really in the clutches of the infinitely small;
for we imply the notion of h being infinitely
near to a. This is just what we want to get
rid of.
Accordingly, we shall yet again restate our
phrase to be explained, and ask what we
mean by saying that the limit of the function
(fh) at a is 1.
The limit of f(h) at a is a property of the
neighbourhood of a, where "neighbourhood"
is used in the sense defined in Chapter XI
during the discussion of the continuity of
functions. The value of the function f(h) at
a is f(a); but the limit is distinct in idea
from the value, and may be different from
it, and may exist when the value has not
been defined. We shall also use the term
"standard of approximation" in the sense
in which it is defined in Chapter XI. In
fact, in the definition of "continuity" given
towards the end of that chapter we have
practically defined a limit. The definition of
a limit is: —
A function f(x) has the limit I at a value
a of its argument x, when in the neighbourhood of a its values approximate to l within
every standard of approximation.
Compare this definition with that already
given for continuity, namely: