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

DIFFERENTIAL CALCULUS 23S
in the light of our definition of a limit. We
have
(x +h)2 _-2 2hx +h2 h(2x +h)
h h h
Now in finding the limit of h(2 +h) at the
h
value O of the argument h, the value (if any)
of the function at h =O is excluded. But for
all values of h, except h=O, we can divide
through by h. Thus the limit of h(x +h)at
h at
h=O is the same as that of 2x+h at h=O.
Now, whatever standard of approximation k
we choose to take, by considering the interval
from -~k to +~k we see that, for values of h
which fall within it, 2x+h differs from 2x by
less than kk, that is by less than k. This is true
for any standard k. Hence in the neighbourhood of the value O for h, 2x +h approximates
to 2x within every standard of approximation, and therefore 2x is the limit of 2x+h
at h = O. Hence by what has been said above
2x is the limit of (x+h)2-x at the value O
h
for h. It follows, therefore, that 2x is what
we have called the rate of increase of x2 at
the value x of the argument. Thus this
method conducts us to the same rate of in