Calculus: Analytic Geometry and Calculus, with Vectors

124 Functions, limits, derivatives

same as that of the assertion

(3.22)

To each positive number e there corresponds

a positive number S such that

Ix2- 91 < e whenever 0 < Ix - 31 < S

which we shall call the epsilon-delta assertion.t When we first see the

epsilon-delta assertion, we are entitled to feel that it lacks the intuitional

appeal of the preceding assertions, but it turns out to be the fully mean-

ingful assertion which can be proved if it is true and can be disproved if it

is false.

Before further discussion of the assertions in the boxes, we can note

that they are so long that it is tedious to write them very often and that

they are universally abbreviated by the efficient and effective shorthand

(3.23) Jim x2 = 9X- s

in our fourth and final box. Thus the assertions in the four boxes are

equivalent; if one is true, then all four are true; and if one is false, then all

four are false. They all mean the same thing.

The only possible discordant phrase in the symphony is the noise we

make when we read the assertion in the last box. We say that the limit

as x approaches 3 of x2 is 9. Thus we have another technical statement

couched in terms of the dubious concept of moving numbers. Stephen

Leacock (1869-1944) was wise enough to realize that if a number x really

could approach 3 from more than one direction, then it should be able to

reverse the process and go away from 3 in more than one direction. In

any case, Leacock enabled Lord Ronald (a character in "Nonsense Novels,

Gertrude the Governess: or, Simple Seventeen") to fling himself upon his

horse and ride madly off in all directions. We make no attempt to explain

our basic concept in terms of moving numbers. Such attempts are much

too mystic and vague for advanced technical books, and we can hold the

view that they are at least a little bit.too mystic and vague for elementary

books. In our book, the collection of words "the limit as x approaches

3 of x2 is 9" does not suggest that numbers jump around; it suggests that

"xt is near 9 whenever x is near 3 but x is different from 3," and this

basic concept is made precise by the epsilon-delta assertion.

f In this and similar assertions, we avoid difficulties by using the word "each" in prefer-

ence to "any" because the troublesome word "any" often means "at least one." Residents

of Los Angeles can be expected to give lusty affirmative answers when asked whether any

major city of the United States lies west of the Mississippi River. If ei < es and Jx2 - 91 <

el, then J x2 - 91 < se. For this reason the epsilon-delta assertion will be true if to each a for

which 0 < e < 0.001 there corresponds a positive number S such that Jx2- 91 < e

whenever 0 < Ix - 31 < S.