The multiplicative identity of R[ x] is the polynomial x 0; that is, x 0 times any polynomial p( x) is just p( x).

Beginning of the discussion about the power functions for the revision of the IEEE 754 standard, May 2007. He deduced that the limit of the full two-variable function x y without a specified constraint is "indeterminate". In 1752, Euler in Introductio in analysin infinitorum wrote that a 0 = 1 [14] and explicitly mentioned that 0 0 = 1. 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. 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.

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. 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. Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Suggestion of variants in the discussion about the power functions for the revision of the IEEE 754 standard, May 2007. 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. Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form. According to Benson (1999), "The choice whether to define 0 0 is based on convenience, not on correctness. 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".We must define x 0 = 1, for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = − y. 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). Möbius reduced to the case c = 0, but then made the mistake of assuming that each of f and g could be expressed in the form Px n for some continuous function P not vanishing at 0 and some nonnegative integer n, which is true for analytic functions, but not in general.

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.On the other hand, if f and g are analytic functions on an open neighborhood of a number c, then f( t) g( t) → 1 as t approaches c from any side on which f is positive. 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).

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). 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. Euler, when setting 0 0 = 1, mentioned that consequently the values of the function 0 x take a "huge jump", from ∞ for x< 0, to 1 at x = 0, to 0 for x> 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.

More precisely, for any given real number r, there is a unique unital R-algebra homomorphism ev r: R[ x] → R such that ev r( x) = r. There is also the exponentiation operator