3.2 Limits 125It is both easy and customary to adopt the absurd view that everybody
has spent huge amounts of time squaring all sorts of numbers near 3 and
has somehow picked up positive knowledge that the assertions in the
boxes are true. Instead of trying to discover how uncertain we should be,
we eliminate uncertainties by proving the epsilon-delta assertion.
Let e > 0. If 0 < S < 1 and 3 < e/7, then
(3.24) Ix2 - 91 = I (x + 3) (x - 3)j
=Ix+31lx-31 <71x-31 <76 <whenever 0 < Ix - 31 < 6. To obtain the inequality ix + 31 < 7
which was used in (3.24), we can use an appropriate figure or, alterna-
tively, use the fact that if Ix - 31 < S and 5 < 1, then
Ix+31 =Ix -3+615Ix -3(+6<5+6<7.
Thus an appropriate 6 can be found and the assertion is true. It is not
inappropriate to think about this matter for a few minutes or perhaps
longer.f
We are familiar with the nature of the graph
of the equation y = x2, and it is comforting to 9+
see that the epsilon-delta assertion has a simple
geometric interpretation. It says that if e > 0^9
and if horizontal lines are drawn through the s-e
i
11points with y coordinates 9 - e and 9 + a as in
Figure 3.241, then there exist vertical lines
(dotted in the figure) such that, with the possible
exception of a single point for which x = 3, the
part of the graph between the vertical lines is also
between the horizontal lines. The little sister we
mention occasionally might be irked by the pos-
sible exceptional point, but she certainly would 3-63+5
be clever enough to put in the dotted lines after Figure 3.241
we had shown her a figure containing the hori-
zontal lines; the process is thoroughly elementary and we need not
require that efforts be made to seek the greatest 6 that serves the pur-
pose. Even though it does not make precise mathematical sense to say
t The famous flea assertion "each flea has a smaller flea to bite him" is, in some respects,
similar to the epsilon-delta assertion. We recognize that the "each flea" at the front of the
assertion invites us to think about fleas one at a time, not every flea or all fleas at once. To
prove the flea assertion, we would be required to start with a given flea, say Mr. F. (who
could be any flea but would not be every flea or the collection of all fleas), and show that
there is a smaller flea which is so related to Mr. F. (and which may be said to correspond
to Mr. F.) that it bites him. To prove the epsilon-delta assertion, it suffices to start with a
given positive number e (which could be 416 or or 0.00001 or any other positive number
but naturally cannot be all of these things at once) and then show that there is a positive
number 6 so related to e that jx2 - 91 < e whenever x 0 3 and Ix - 31 < S.