section_path: Limits of powers
section_name: Limits of powers
paragraph_text: Limits of powers
  
  Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 0. The limits in these examples exist, but have different values, showing that the two-variable function x has no limit at the point (0, 0). One may consider at what points this function does have a limit.
  
  More precisely, consider the function f (x, y) = x y {\displaystyle f(x,y)=x^{y}} defined on D = { (x, y) ∈ R 2: x > 0 } {\displaystyle D=\{(x,y)\in \mathbf {R} ^{2}:x>0\}}. Then D can be viewed as a subset of R (that is, the set of all pairs (x, y) with x, y belonging to the extended real number line R =, endowed with the product topology), which will contain the points at which the function f has a limit.
  
  In fact, f has a limit at all accumulation points of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞). Accordingly, this allows one to define the powers x by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 0, (+∞), 1 and 1, which remain indeterminate forms.
  
  Under this definition by continuity, we obtain:
  
  These powers are obtained by taking limits of x for positive values of x. This method does not permit a definition of x when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D.
  
  On the other hand, when n is an integer, the power x is already meaningful for all values of x, including negative ones. This may make the definition 0 = +∞ obtained above for negative n problematic when n is odd, since in this case x → +∞ as x tends to 0 through positive values, but not negative ones.