4.3 THE LIMIT CONCEPT (OPTIONAL) 135Y
Figure 4.4 A graphic indication that
1 # lim,,, f (x). If - 6 < x c 0, then f (x) does
not fall between 0 and 2, no matter how small we
choose 6.
positive 6 be given as shown in Figure 4.4. Now, can we find a value
of x between 0 - 6 and 0 + 6 (along the x axis) for which f(x) is
between 0 and 2 (along the y axis)? We can indeed! Simply choose any x
between -6 and 0, that is, any x in the left half of the interval (-6, S),
and note that f(x) equals x - 1, which is less than - 1, and hence is not
between 0 and 2.
Note that the key to the preceding argument was the choice of e small
enough so that, within any vertical 6 band about the line x = 0, no
matter how narrow, at least part of the graph off, inside that vertical
band, does not lie inside the horizontal e band surrounding the line
y = 1 (y = L in general). What values of e, other than e = 1, would have
been permissible? Surely, any positive e less than a value that is known
to "work," as e = 1 does in the preceding argument, will also work.
(Think about this statement for a moment and make sure you under-
stand it.) So the question is how large a value of E will permit us to use
successfully the preceding argument. Could we, for instance, let E = $,
giving rise-to an band if width 5 about the line y = l? See Figure 4.5.
Again, let an arbitrary positive 6 be chosen. Examine the picture in this
case; it is clear from the picture that if 6 is reasonably small, say, 4,
we cannot find a value of x within 6 of x = 0 for which f(x) lies outside
the given E band. So e = 1 works, whereas E = $ is too large. Where is
the dividing line between values of epsilon that work and those that
don't? Geometric sense tells us that it is E = 2, the vertical distance
between the proposed limit (L = 1 in this case) and the part of the graph