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

232 INTRODUCTION TO MATHEMATICS
meaning. But for every other value of x,
2Or
the value of the function x is 2. Thus the
x
limit of x at x=0 is 2, and it has no value
x
x2
at x =0. Similarly the limit of - at x=a is
x
a whatever a may be, so that the limit of
52 X2
- at x=0 is 0. But the value of at x=0
x x
takes the form 0-, which has no defined
meaning. Thus the function - has a limit
but no value at 0.
We now come back to the problem from
which we started this discussion on the nature
of a limit. How are we going to define the
rate of increase of the function x2 at any
value x of its argument. Our answer is that
this rate of increase is the limit of the function (x +h)2 -2 at the value zero for its
h
argument h. (Note that x is here a "constant." Let us see how this answer works