Calculus textbooks also discuss the problem, usually in a section dealing with L'Hospital's Rule. Suppose we are given two functions, *f*(*x*) and *g*(*x*), with the properties that \(\lim_{x\rightarrow a} f(x)=0\) and \(\lim_{x\rightarrow a} g(x)=0.\) When attempting to evaluate [*f*(*x*)]^{g(x)} in the limit as *x* approaches *a*, we are told rightly that this is an *indeterminate form* of type 0^{0} and that the limit can have various values of *f* and *g*. This begs the question: are these the same? Can we distinguish 0^{0} as an indeterminate form and 0^{0} as a number?

The treatment of 0^{0} has been discussed for several hundred years. Donald Knuth [7] points out that an Italian count by the name of Guglielmo Libri published several papers in the 1830s on the subject of 0^{0} and its properties. However, in his *Elements of Algebra*, (1770) [4], which was published years before Libri, Euler wrote,

As in this series of powers each term is found by multiplying the preceding term by

a,which increases the exponent by 1; so when any term is given, we may also find the preceding term, if we divide bya,because this diminishes the exponent by 1. This shews that the term which precedes the first terma^{1}must necessarily bea/aor 1; and, if we proceed according to the exponents, we immediately conclude, that the term which precedes the first must bea^{0}; and hence we deduce this remarkable property, thata^{0}is always equal to 1, however great or small the value of the numberamay be, and even when a is nothing; that is to say,a^{0}is equal to 1.

More from Euler: In his *Introduction to Analysis of the Infinite* (1748) [5], he writes :

Let the exponential to be considered beawhere a is a constant and the exponent^{z}zis a variable .... Ifz= 0, then we have a^{0}= 1. Ifa= 0, we take a huge jump in the values ofa. As long as the value of^{z}zremains positive, or greater than zero, then we always havea= 0. If^{z}z= 0, thena^{0}= 1.

Euler defines the logarithm of *y* as the value of the function *z,* such that *a ^{z}* =

*Editor's note:* *This article was published in March of 2008.*