Exponentiation - Limits of Powers

Limits of Powers

The section zero to the power of zero gives a number of examples of limits which are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function xy has no limit at the point (0,0). One may ask at what points this function does have a limit.

More precisely, consider the function f(x,y) = xy defined on D = {(x,y) ∈ R2 : x > 0}. Then D can be viewed as a subset of R2 (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 xy by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.

Under this definition by continuity, we obtain:

• x+∞ = +∞ and x−∞ = 0, when 1 < x ≤ +∞.
• x+∞ = 0 and x−∞ = +∞, when 0 ≤ x < 1.
• 0y = 0 and (+∞)y = +∞, when 0 < y ≤ +∞.
• 0y = +∞ and (+∞)y = 0, when −∞ ≤ y < 0.

These powers are obtained by taking limits of xy for positive values of x. This method does not permit a definition of xy 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 xn is already meaningful for all values of x, including negative ones. This may make the definition 0n = +∞ obtained above for negative n problematic when n is odd, since in this case xn → +∞ as x tends to 0 through positive values, but not negative ones.