Calculus: Analytic Geometry and Calculus, with Vectors

3.2 Limits 123

involve only simple words and may seem, at first sight, to be childishly
simple.
It is reasonable to suppose that the harangue of the previous paragraph

is leading up to something, and that the lightning is about to strike. It

is. We are going to undertake to make a sane appraisal of the assertion

(3.21) x2 is near 9 whenever x is near 3 but x 5,16 3

which we shall call the assertion in the first box. The assertion does not
say anything about the value of x2 when x = 3. It does not say that
x2 is 9 when x = 3, and hence it does not pretend to tell the whole truth.
There is a fundamental reason why it is not completely easy to tell what
the assertion does mean. The reason is that it simply does not make
precise mathematical sense to say that a number x is near 3. Whether
416 or 4 or 3.01 or 3.00001 or 2.98 is considered to be near 3 or not can be a
matter of opinion and can depend upon circumstances. Likewise, it
does not make precise mathematical sense to say that x2 is near 9.
Discouraging as this may be, we must recognize that it may be possible
to attach a precise meaning to the assertion in the first box without attach-
ing meanings to the "assertions" x is near 3 and x2 is near 9. After all,
the word "attaching" can mean something even when "atta" and
"ching" do not. It should be possible to tell precisely what the assertion
does mean, because the assertion uses words in a thoroughly serious
attempt to convey information. A fundamental idea is involved.
Our first attempt to make sense out of (3.21) is to replace it by the
assertion

(3.211)

x2 is a good approximation to 9 whenever x is

a good approximation to 3 but x 0 3

in the second box. This change in the wording can be psychologically

satisfying, and we started with (3.21) only because it is shorter than

(3.211). We have not, however, conquered our fundamental difficulty,

because the statement that one number is a good approximation to

another is neither more nor less illuminating than the statement that one

is near the other.

It is a remarkable fact that much of the mathematical progress of the

past century is based upon the development and use of a particular special

method of attaching meaning to the statements in the first two boxes.

The method is called the epsilon-delta method because it traditionally

employs the two Greek letters a (epsilon) and S (delta). The meaning of

the assertions in the first two boxes is, by this method, defined to be the