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What is 0^0? - Indeterminate Forms

Michael Huber and V. Frederick Rickey


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 az 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 az. As long as the value of z remains positive, or greater than zero, then we always have az = 0. If z = 0, then a0 = 1.

Euler defines the logarithm of y as the value of the function z, such that az = 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.

Michael Huber and V. Frederick Rickey, "What is 0^0? - Indeterminate Forms," Convergence (July 2012)