The pown and powr variants have been introduced due to conflicting usage of the power functions and the different points of view (as stated above). In general the limit of φ( x)/ ψ( x) when x = a in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. Zero to the power of zero, denoted by 0 0, is a mathematical expression that is either defined as 1 or left undefined, depending on context. On the other hand, in 1821 Cauchy [20] explained why the limit of x y as positive numbers x and y approach 0 while being constrained by some fixed relation could be made to assume any value between 0 and ∞ by choosing the relation appropriately.

The combinatorial interpretation of b 0 is the number of 0-tuples of elements from a b-element set; there is exactly one 0-tuple. With this justification, he listed 0 0 along with expressions like 0 / 0 in a table of indeterminate forms. In the 1830s, Libri [18] [16] published several further arguments attempting to justify the claim 0 0 = 1, though these were far from convincing, even by standards of rigor at the time. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents.But for C, as of C99, if the normative annexF is supported, the result for real floating-point types is required to be 1 because there are significant applications for which this value is more useful than NaN [28] (for instance, with discrete exponents); the result on complex types is not specified, even if the informative annexG is supported. Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library; this is the case with Lua [33] and Perl's ** operator [34] (where it is explicitly mentioned that the result of 0**0 is platform-dependent).

He deduced that the limit of the full two-variable function x y without a specified constraint is "indeterminate".Some textbooks leave the quantity 0 0 undefined, because the functions x 0 and 0 x have different limiting values when x decreases to 0. 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.

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. Suggestion of variants in the discussion about the power functions for the revision of the IEEE 754 standard, May 2007. Apparently unaware of Cauchy's work, Möbius [8] in 1834, building on Pfaff's argument, claimed incorrectly that f( x) g( x) → 1 whenever f( x), g( x) → 0 as x approaches a number c (presumably f is assumed positive away from c). 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). This does not define 0 w since there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.The set-theoretic interpretation of b 0 is the number of functions from the empty set to a b-element set; there is exactly one such function, namely, the empty function. The multiplicative identity of R[ x] is the polynomial x 0; that is, x 0 times any polynomial p( x) is just p( x).