198 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4
Figure 4.5 Not just any E can be used in a proof that
L # lim,,, f(x). The E band (E = $1 - 3 < y < S is too
wide to be us@ in proving 1 # lim f(x) as x tends to 0.
If 6 = i, then euery x within 6 of x = 0 has its
corresponding f(x) between -9 and 3, that is, within the
given E band.whose y values do not all "lie close to" L (the part of the graph off to
the left of x = 0 in this case). Thus, in the argument that 1 # lirn,,, f (x),
given in the previous paragraph, we could have used as our E any specific
positive number less than or equal to 2, rather than E = 1, with the
same argument working.
We next consider how that argument would have to be modified in
order to show that - 1 # lim,,, f(x). Again, we could start by specify-
ing E = 1 (or E equals any other specific positive number less than or
equal to 2). This time, for an arbitrarily' chosen positive S, we can find
values of x to the right of x = 0 whose functional value f(x) = 2x + 1,
being greater than 1, is certainly not within distance E = 1 of the pro-
posed limit L = - 1. Hence this value of L fails to be a limit as x
approaches 0.
L = 0, being halfivay between the two separate pieces of the graph,
might be thought to serve as lirn,,, f(x). How can we use an epsilon-
delta argument to discredit this idea? We must specify a value of E; can
we still use any value as large as 2? See Figure 4.6. The answer is "no!"
Since the vertical distance from L = 0 to the part of the graph on either
side of x = 0 is 1, we must start with a specific E less than or equal to
1, say, E = t. YOU should try to complete the argument that, for any
positive 6, no matter how small, there can always be found a value of
x within 6 of x = 0, whose corresponding f(x) is not within a distance
c = $ of L = 0. Also, determine the largest value of E that can be used
to prove that L # lirn,,, f (x), where L = 4, L = 3, L = - 5.
Finally, if ambitious, you may want to consider the problem of argu-
ing that L # lirn,,, f(x), where L is an arbitrary number along the y