What is the Undefined Value of 1^∞?

[dissolver]

Why is 1^ \infty. undefined?

 

[StatusX]

This has been discussed here recently. The gist is that 1^\infty can only be interpreted as standing for a class of limit expressions, including the following:

\lim_{x \rightarrow \infty} 1^x = 1

\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = e

\lim_{x \rightarrow \infty} (1+\frac{1}{x})^{x^2} = \infty

Since these don't all have the same limit (as opposed to, say, the limit expressions corresponding to 0/\infty), we can't assign a value to 1^\infty.