Why not? The limit of x y along the line x = 0 is 0, but the limit along the line y = 0 is 1, not 0. No matter what value may be assigned to 0 0, the function x y can never be continuous at x = y = 0. When might a mathematician want 0 0 to be something that is not indeterminate? If, for example, we are discussing the function f( x, y) = x y, the origin is a discontinuity of the function.
'So, as far as cardinal numbers are concerned,' wrote Vaughan, '0 0 = 1.' There is precisely one such mapping, which is itself the set of the empty set. In order to calculate 0 0, determine the number of mappings of the empty set into itself. Suppose we are given two functions, f( x) and g( x), with the properties that \(\lim_. Calculus textbooks also discuss the problem, usually in a section dealing with L'Hospital's Rule.
