# What is infinity to the power of zero?

1. May 12, 2012

1. The problem statement, all variables and given/known data
Hi everyone,

I'm just wondering if someone could please clarify for me what infinity to the power of zero is? I seem to be finding conflicting opinions about this online. Is it '1' or 'not defined'?

Thanks!

2. Relevant equations

3. The attempt at a solution

2. May 12, 2012

### Steely Dan

Infinity is not a legitimate number with which you could ask what the result of raising it to the zero power is.

3. May 12, 2012

Thanks!

4. May 13, 2012

### HallsofIvy

Staff Emeritus
More specifically, "infinity" is not a member of the real number system on whicy our standard operations are defined. There do exist "extended" number systems in which "infinity" is defined but then the usual arithmetic operations to not apply.

5. May 13, 2012

### Mentallic

It's undefined. Also we never say what infinite to the power of zero is, we instead express such results by the use of limits:

$$\lim_{x\to \infty}x^{1/x}$$ is such an expression that would be of the form $\infty ^0$ and in this case it's equal to 1, but there are many other cases where it's not, such as $$\lim_{x\to \infty}\left(x^x\right)^{1/x}=\infty$$ or $$\lim_{x\to \infty} x^{1/\ln(x)}$$ which is a special one that equals the irrational number $e\approx 2.718$

This is why expressions of this form are called indeterminate. They're undefined, until you find the specific question that defines them. It's different from the sense of 1/0 being undefined since that one is undefinable since it would create inconsistencies in our maths.