What is 0^0? - Indeterminate Forms

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 0⁰ and that the limit can have various values of f and g. This begs the question: are these the same? Can we distinguish 0⁰ as an indeterminate form and 0⁰ as a number?

The treatment of 0⁰ 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 0⁰ 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 a¹ 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 a⁰; and hence we deduce this remarkable property, that a⁰ 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, a⁰ 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 a⁰ = 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 a⁰ = 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 0⁰.

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