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".

A (real) polynomial is an expression of the form a 0 x 0 + ⋅⋅⋅ + a n x n, where x is an indeterminate, and the coefficients a i are real numbers.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). 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).

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.The multiplicative identity of R[ x] is the polynomial x 0; that is, x 0 times any polynomial p( x) is just p( x). There is also the exponentiation operator 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. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. 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.