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

DIFFERENTIAL CALCULUS 231
value 3 of the argument x, but by definition
f(3) =0.
This is an instance of a function which
possesses both a value and a limit at the
value 3 of the argument, but the value is not
equal to the limit. At the end of Chapter
XI the function x2 was considered at the
value 2 of the argument. Its value at 2 is 22,
i.e. 4, and it was proved that its limit is also
4. Thus here we have a function with a
value and a limit which are equal.
Finally we core to the case which is essentially important for our purposes, namely, to
a function which possesses a limit, but no
defined value at a certain value of its argument. We need not go far to look for
2x
such a function, - will serve our purpose.
x
Now in any mathematical book, we might
find the equation, = 2, written without
x
hesitation or comment. But there is a difficulty in this; for when x is zero, - =; and
x 0
lhas no defined meaning. Thus the value
o0 ~2x
of the function at x =0 has no defined
x