Can exponents ever be infinity? In some non-mathematics textbooks, authors would define a quantity $x^y$, where $x$ and $y$ are some variables, and it is possible for $y = \infty$. The variable $y$ can take on the value $\infty$, not just in the limit.
So, I assume they are working in the extended real line. However, for $x \in \mathbb{R}$, can we have $x^{\infty}$ in the extended real line? If not, could they be using a different number system for which this is valid?
E.g. if $0 < x < 1$, they would say $x^{\infty} = 0$. Not in the sense of the limit. Is this valid?

The function $f(x,y) = x^y$ is continuous for $x \geq 0$ and $y \in \mathbb{R}$. It would be a shame to ruin that by letting $y \in \mathbb{R}\cup\{\infty\}$, so presumably we would want $x^y$ to be continuous for $y$ in the extended reals as well.
This is precisely the statement that $\lim_{y \to \infty} x^y = x^\infty$. So if we were to define such a function, the only reasonable definition would be fore it to be in the limit sense.
