Zero to the Power of Zero/As a Limit

Example of Zero to the Power of Zero

Consider the real function:

$y = x^x$

This function is well defined for $x > 0$.

It is not obvious whether or not the right hand limit:

$\ds \lim_{x \mathop \to 0^+} y$

exists.

If it does, it would be nice if:

$\ds \lim_{x \mathop \to 0^+} x^x = 0^0$

Indeed, by Limit of x to the x, we have:

$\ds \lim_{x \mathop \to 0^+} x^x = 1$

We see that defining $0^0 = 1$ allows $x^x$ to be right-continuous at $x = 0$.