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

228 INTRODUCTION TO MATHEMATICS
saying that 2x+h is 2x, when h is zero. But
this will not do; for it thereby abolishes the
interval from x to x +h, over which the average increase was calculated. The problem is,
how to keep an interval of length h over which
to calculate the average increase, and at the
same time to treat h as if it were zero. Newton did this by the conception of a limit, and
we now proceed to give Weierstrass's explanation of its real meaning.
In the first place notice that, in discussing
2x +h, we have been considering x as fixed in
value and h as varying. In other words x
has been treated as a "constant" variable,
or parameter, as explained in Chapter IX;
and we have really been considering 2x+h as
a function of the argument h. Hence we can
generalize the question on hand, and ask
what we mean by saying that the function
f(h) tends to the limit 1, say, as its argument
h tends to the value zero. But again we shall
see that the special value zero for the argument
does not belong to the essence of the subject;
and again we generalize still further, and ask
what we mean by saying that the functionf(h)
tends to the limit I as h tends to the value a.
Now, according to the Weierstrassian explanation the whole idea of h tending to the
value a, though it gives a sort of metaphorical
picture of what we are driving at, is really off
the point entirely. Indeed it is fairly obvious