According to some Calculus textbooks, *0^0* is an ``indeterminate
form''. When evaluating a limit of the form *0^0*, then you need
to know that limits of that form are called ``indeterminate forms'',
and that you need to use a special technique such as L'Hopital's
rule to evaluate them. Otherwise, *0^0 = 1* seems to be the most
useful choice for *0^0*. This convention allows us to extend
definitions in different areas of mathematics that otherwise would
require treating 0 as a special case. Notice that *0^0* is a
discontinuity of the function *x^y*. More importantly, keep
in mind that the value of a function and its limit need not be the
same thing.

This means that depending on the context where *0^0* occurs, you
might wish to substitute it with 1, indeterminate or
undefined/nonexistent.

Some people feel that giving a value to a function with an
essential discontinuity at a point, such as *x^y* at *(0,0)*, is
an inelegant patch and should not be done. Others point out
correctly that in mathematics, usefulness and consistency are
very important, and that under these parameters *0^0 = 1* is
the natural choice.

The following is a list of reasons why *0^0* should be 1.

Rotando & Korn show that if *f* and *g* are real functions that vanish
at the origin and are analytic at 0 (infinitely differentiable is
not sufficient), then *f(x)^(g(x))* approaches 1 as *x* approaches 0 from
the right.

From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):

Some textbooks leave the quantityPublished by Addison-Wesley, 2nd printing Dec, 1988.0^0undefined, because the functionsx^0and0^xhave different limiting values whenxdecreases to 0. But this is a mistake. We must definex^0 = 1for allx, if the binomial theorem is to be valid whenx=0,y=0, and/orx=-y. The theorem is too important to be arbitrarily restricted! By contrast, the function0^xis quite unimportant.

As a rule of thumb, one can say that *0^0 = 1*, but *0.0^(0.0)* is
undefined, meaning that when approaching from a different
direction there is no clearly predetermined value to assign to
*0.0^(0.0)*; but Kahan has argued that *0.0^(0.0)* should be 1,
because if *f(x), g(x) -> 0* as *x* approaches some
limit, and *f(x)* and *g(x)* are analytic functions, then
*f(x)^g(x) -> 1*.

The discussion on *0^0* is very old, Euler argues for *0^0 = 1*
since *a^0 = 1* for *a != 0*. The controversy raged throughout the
nineteenth century, but was mainly conducted in the pages of the
lesser journals: Grunert's Archiv and Schlomilch's
Zeitshrift. Consensus has recently been built around setting the
value of *0^0 = 1*.

On a discussion of the use of the function *0^(0^x)* by an Italian
mathematician named Guglielmo Libri.

[T]he paper [33] did produce several ripples in mathematical waters when it originally appeared, because it stirred up a controversy about whether0^0is defined. Most mathematicians agreed that0^0 = 1, but Cauchy [5, page 70] had listed0^0together with other expressions like0/0andoo - ooin a table of undefined forms. Libri's justification for the equation0^0 = 1was far from convincing, and a commentator who signed his name simply ``S'' rose to the attack [45]. August Möbius [36] defended Libri, by presenting his former professor's reason for believing that0^0 = 1(basically a proof thatlim_(x -> 0+) x^x = 1). Möbius also went further and presented a supposed proof thatlim_(x -> 0+) f(x)^(g(x))wheneverlim_(x -> 0+) f(x) = lim_(x -> 0+) g(x) = 0. Of course ``S'' then asked [3] whether Möbius knew about functions such asf(x) = e^(-1/x)andg(x) = x. (And paper [36] was quietly omitted from the historical record when the collected words of Möbius were ultimately published.) The debate stopped there, apparently with the conclusion that0^0should be undefined.But no, no, ten thousand times no! Anybody who wants the binomial theorem

(x + y)^n = sum_(k = 0)^n (nk) x^k y^(n - k)to hold for at least one nonnegative integernmustbelieve that0^0 = 1, for we can plug inx = 0andy = 1to get 1 on the left and0^0on the right.The number of mappings from the empty set to the empty set is

0^0. Ithasto be 1.On the other hand, Cauchy had good reason to consider

0^0as an undefinedlimiting form, in the sense that the limiting value off(x)^(g(x))is not knowna prioriwhenf(x)andg(x)approach 0 independently. In this much stronger sense, the value of0^0is less defined than, say, the value of0+0. Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side.[3]

Anonymous and S...Bemerkungen zu den Aufsatze überschrieben, `Beweis der Gleichung0^0 = 1, nach J. F. Pfaff',im zweiten Hefte dieses Bandes, S. 134, Journal für die reine und angewandte Mathematik,12 (1834), 292-294.

[5]

Ouvres Complètes.Augustin-Louis Cauchy.Cours d'Analyse de l'Ecole Royale Polytechnique (1821). Series 2, volume 3.

[33]

Guillaume Libri.Mémoire sur les fonctions discontinues, Journal für die reine und angewandte Mathematik,10 (1833), 303-316.

[36]

A. F. Möbius.Beweis der Gleichung0^0 = 1, nach J. F. Pfaff.Journal für die reine und angewandte Mathematik,12 (1834), 134-136.

[45]

S...Sur la valeur de0^0.Journal für die reine und angewandte Mathematik 11,(1834), 272-273.

**References**

*Knuth.* **Two notes on notation.** *(AMM 99 no. 5 (May 1992),* 403-422).

*H. E. Vaughan.* **The expression ' 0^0'.**

*Louis M. Rotando and Henry Korn.* **The Indeterminate Form 0^0.**

*L. J. Paige,.* **A note on indeterminate forms.** *American
Mathematical
Monthly,* 61 (1954), 189-190; reprinted in the Mathematical
Association of America's 1969 volume, Selected Papers on Calculus,
pp. 210-211.

*Baxley & Hayashi.* **A note on indeterminate
forms.** *American Mathematical
Monthly,* 85 (1978), pp. 484-486.

Sat Jan 6 22:47:26 EST 1996