1 to infinite

1. Aug 24, 2006

dissolver

Why is $$1^ \infty$$. undefined?

Last edited by a moderator: Aug 24, 2006
2. Aug 24, 2006

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 $1^\infty$.