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 $1^\infty$.

 

[tacman]

The undefined value of 1^∞ is a mathematical concept that means the result of raising 1 to the power of infinity is not defined or cannot be determined. This is because infinity is not a number, it is a concept that represents something that is unbounded or limitless. When we raise a number to the power of infinity, we are essentially asking what number multiplied by itself an infinite number of times will give us the original number. This concept is undefined because it is impossible to determine the exact value of such a calculation. Additionally, as the exponent increases towards infinity, the result of 1^∞ approaches either 0 or infinity, depending on the context and mathematical rules being applied. Therefore, 1^∞ is undefined because the result is not a specific number and cannot be accurately calculated.