Calculus: Analytic Geometry and Calculus, with Vectors

720 Appendix I

of ham (or a fly) is between them, then the thing that is caught in the
middle must also be near Minneapolis. To prove this theorem, let
e > 0. Choose S such that 0 < b < p and the two inequalities

(37) L-e < g(x) <L+e, L-e <h(x) <L+e

hold when 0 < Ix - al < S. This and (34) give

(38) L - e < g(x) < f(x) S h(x) < L + e

and hence

(39) If(x) - Lj < e

when 0 < Ix - al < S. This proves Theorem G.

Theorem 3.288, the last one of the basic theorems of Section 3.2,

asserts that if p is a constant positive exponent and a >_ 0, then

limx2 = a2.

x-a

Proof of this theorem is much more difficult and is given in Section 9.2

after the theory of exponentials and logarithms has been developed;

see Theorem 9.271.

We conclude this appendix with an indication of the extent to which

mathematical fashions have changed. In a calculus textbook published

in 1879 and cited in a footnote near the end of Chapter 3, W. E. Byerly

says he "embodies the results of my own experience in teaching the

calculus at Cornell and Harvard Universities." His preface claims that

one of the "peculiarities" of his book is "rigorous use of the Doctrine of

Limits as a foundation of the subject." His basic definition of limit

appears on page 3. "If a variable which changes its value according to

some law can be made to approach some fixed, constant value as nearly as

we please, but can never become equal to it, the constant is called the limit

of the variable under the circumstances in question." The "fundamental

proposition" in the theory of limits is given as a theorem on page 5:

THEOREM. If two variables are so related that as they change they keep always

equal to each other, and each approaches a limit, their limits are absolutely equal.

For two variables so related that they are always equal form but a single

varying value, as at any instant of their change they are by hypothesis absolutely

the same. A single varying value cannot be made to approach at the same time

two different constant values as nearly as we please; for, if it could, it could

eventually be made to assume a value between the two constants; and, after

that, in approaching one it would recede from the other.

This appendix is based upon the premise that such "definitions" and

"proofs" outlived their usefulness as their staunch defenders insisted

that it is easier to learn them than to learn definitions and proofs involv-

ing epsilons and deltas.