Cauchy, Augustin-Louis (1821), Cours d'Analyse de l'École Royale Polytechnique, Oeuvres Complètes: 2 (in French), vol. In the complex domain, the function z w may be defined for nonzero z by choosing a branch of log z and defining z w as e w log z. The expression 0 0 is an indeterminate form: Given real-valued functions f( t) and g( t) approaching 0 (as t approaches a real number or ±∞) with f( t) > 0, the limit of f( t) g( t) can be any non-negative real number or +∞, or it can diverge, depending on f and g.

An anonymous commentator pointed out the unjustified step; [21] then another commentator who signed his name simply as "S" provided the explicit counterexamples ( e −1/ x) x → e −1 and ( e −1/ x) 2 x → e −2 as x → 0 + and expressed the situation by writing that " 0 0 can have many different values". Thus, the two-variable function x y, though continuous on the set {( x, y): x> 0}, cannot be extended to a continuous function on {( x, y): x> 0} ∪ {(0, 0)}, no matter how one chooses to define 0 0. Other authors leave 0 0 undefined because 0 0 is an indeterminate form: f( t), g( t) → 0 does not imply f( t) g( t) → 1. He deduced that the limit of the full two-variable function x y without a specified constraint is "indeterminate".Knuth (1992) contends more strongly that 0 0 " has to be 1"; he draws a distinction between the value 0 0, which should equal 1, and the limiting form 0 0 (an abbreviation for a limit of f( t) g( t) where f( t), g( t) → 0), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side.

Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. With this justification, he listed 0 0 along with expressions like 0 / 0 in a table of indeterminate forms.The consensus is to use the definition 0 0 = 1, although there are textbooks that refrain from defining 0 0. This and more general results can be obtained by studying the limiting behavior of the function ln( f( t) g( t)) = g( t) ln f( t).