The Indeterminate Form 0⚬Author(s): Louis M. Rotando and Henry KornSource: Mathematics Magazine, Vol. 50, No. 1 (Jan., 1977), pp. 41-42Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2689754 .

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lead to unexpected riffling strategies. For instance, to restore a stack of 10 checks to their original order in a minimum number of riffles one should add 4 dummy checks on the top of the stack to give 14 checks, use 4 Ri riffles, and then remove the extra checks.

Riffling theory extends to repeated shuffling of a deck of 52 cards, and, in that context, the riffle R2 is referred to as the "Faro Out Shuffle" by magicians and (some) card players. It is described in detail by Marlo [2]; see also Feller [1], p. 335. With this method of shuffling the original order of a deck of 52 cards is reconstituted after only 8 shuffles - and hence its special interest. However, with riffle Rl (the "Faro In Shuffle") it takes 52 shuffles, and each card passes through every position before return.

This paper was written while M. J. Gardner held a visiting post at Louisiana State University Medical Center. We wish to acknowledge the helpful comments and suggestions of the editors and the referee..

References

[1] W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1, Wiley, New York, 1950, Chapter 15.

[2] E. Marlo, Revolutionary Card Technique, Chicago, 1958, Chapters 6 and 7. [3] S. Brent Morris, The Basic Mathematics of the Faro shuffle, Pi Mu Epsilon J., 6 (Spring 1975) 85-92.

The Indeterminate Form 0?

Louis M. ROTANDO HENRY KORN Westchester Community College

Consider the following limit problems often encountered in elementary calculus textbooks:

lim (sin X )tan x lim (ex+l - e)x

lim (arctan x)x lim (1 - x)sinh(x-l).

In each of the above problems the limit is 1. Curiously enough the limit is also 1 for most similar problems typically included in the exercise sets devoted to indeterminate forms. Yet the limit process does not yield 1 for every example of the type 00: G. C. Watson [1] discusses a generalization of the counterexample

lim a/log x X 0+

and conditions on f wherein limx0o+xfx 1 are investigated by L. J. Paige [2]. The purpose of this note is to further study of the indeterminate form 00 by looking at examples of

the more general form

lim f(x)g(x x 0+

in which f and g are real functions analytic at x = 0, that is, representable there by a Taylor series. In such cases we can show that the limit is 1.

THEOREM. Suppose that f and g are nonzero real analytic functions at x = 0 for which f(x) _ 0 for all positive x sufficiently close to 0. If limx o+ f(x) = limxo+ g (x) = 0, then limx0+ f(x)g(x) 1

VOL. 50, NO. 1, JANUARY 1977 41

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Proof. Applying L'Hospital's Rule, we have

(1) lim f(x)g(x) = exp [lirn log f(x)] = exp [lim f(x)g'(x)l

Since f and g are analytic and approach zero as x approaches zero, it follows by continuity that f(O) = g(O) = 0, and in some neighborhood of x = 0, f and g have the form

f (x) = x 'F(x), F(O) 70

g(x) = x'G(x), G(O) 0

where m and n are positive integers, while F and G are analytic at x = 0. Substituting these into (1) we obtain, after simplification,

(2) 1 t( )g'X' [ 1. ~~~x ' [xF'(x ) + mF(x)] G 2() (2) lim f (x)"(x) - exp [lim x4F() Fx]2x

X-O+ LX-~O+ -F(x)[xG'(x)+ nG(x)]

The numerator N(x) xn[xF'(x) + mF(x)]G2(x) as well as the denominator D(x) -F(x)[xG'(x)+nG(x)] in this last limit are analytic at x=O. Since D(0)X 0, it follows from continuity at x = O that limx-0+ D(x) O 0, and therefore limx0o [N(x)ID(x)] = O. Hence limx-0+f(x)g(x)= 1.

We note in conclusion that a simple linear transformation of variables will permit coverage of the case limx,+f(x)"x. The theorem may also be easily modified to include the case limxo0-f(x)g(x) by requiring f(x) ' 0 for negative x sufficiently close to 0. Finally, if the restriction of "analytic" is weakened to "infinitely differentiable" at x = 0 then the theorem would be false. A nice counterexam- ple to illustrate this is given by letting g(x)= x and

e-,x if xfO f(x) =

O if x=0.

In this case the reader can easily verify that limx0+f(x)g(x) = 0 and limx0-f(x)g(x) = .

The author is indebted to the referee for several suggestions that led to substantial clarification of the proof.

References

[1] G. C. Watson, A note on indeterminate forms, Amer. Math. Monthly, 68 (1961) 490-492. [21 L. J. Paige, A note on indeterminate forms, Amer. Math. Monthly, 61 (1954) 189-190.

Symmetries for Conditioned Ruin Problems

W. A. BEYER

M. S. WATERMAN Los Alamos Scientific Laboratory

In this paper we discuss some interesting symmetries that arise in conditioning random walks on the integers with absorbing boundaries. These results are ancillary to a study done on conditional expected duration of walks used in a mathematical model of cancer tumors [1].

42 MATHEMATICS MAGAZINE

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