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 by *a,* because this diminishes the exponent by 1. This shews that the term which precedes the first term *a*^{1} must necessarily be *a*/*a* or 1; and, if we proceed according to the exponents, we immediately conclude, that the term which precedes the first must be *a*^{0}; and hence we deduce this remarkable property, that *a*^{0} is always equal to 1, however great or small the value of the number *a* may 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 be *a*^{z} where a is a constant and the exponent *z* is a variable .... If *z* = 0, then we have a^{0} = 1. If *a* = 0, we take a huge jump in the values of *a*^{z}. As long as the value of *z* remains positive, or greater than zero, then we always have *a*^{z} = 0. If *z* = 0, then *a*^{0} = 1.

Euler defines the logarithm of *y* as the value of the function *z,* such that *a*^{z} = *y.* He writes that it is understood that the base *a* of the logarithm should be a number greater than 1, thus avoiding his earlier reference to a possible problem with 0^{0}.

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