Calculus: Analytic Geometry and Calculus, with Vectors

128 Functions, limits, derivatives

Theorem 3.287 (sandwich theorem or flyswatter theorem) If for
some positive number p

g (x) < f (X) < h (x)

whena - p <x <aandwhen a <x <a+ p,andif

lim g(x) = L, lim h(x) = L,

z- a

then

lim f (x) = L.

Theorem 3.288 If p is a constant positive exponent, thent the first of

the formulas

lim xP = aP, lim xP = aP

holds when a > 0 and the second holds when a >__ 0.

These theorems are easily understood and will turn out to be very use-

ful. Unless his teacher rules otherwise, each individual student has three

options. He can claim that the theorems are so obvious that they do not

need proof and, even though this is surely a precarious way to start a

successful mathematical career, he may even be right. He can claim that

they are not obviously true but he will accept them because they are

printed and the teacher says there are no misprints. Finally, he may

want to see proof because he is suspicious or inquisitive or wants to

develop abilities to prove things. In the latter case he may attack

Appendix 1 at the end of this book. Whatever we do, we should always

believe that if f(x) lies between g(x) and h(x) and if g(x) and h(x) are both

near L whenever x is near a but x 0 a, then f (x) must be near L whenever

x is near a but x 0 a. This is what the sandwich theorem says, and the

meanings of the other theorems are also simple.

The first two of the following problems are designed to promote under-

standing of the epsilon-delta assertion (3.26). We must always remember

that if the epsilon-delta assertion is true, then to each (not all or every)

epsilon that is positive there corresponds a delta that is positive such that

t f (x) - Lj < ewhenever x is different from a but so near a that Jx- at < 3.

It is not asserted that there is a delta which corresponds to every epsilon.

It is asserted that to each epsilon there corresponds a delta. The epsilon

comes first, and the delta follows.

t The meaning of the second of these statements is explained in Section 3.3. The

theorem is, as Appendix 1 says, proved in Chapter 9 after the theory of exponentials and

logarithms has been developed. See Theorem 9.271.