What is infinity to the power of zero?

[Pyroadept]

Homework Statement

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!

Homework Equations

The Attempt at a Solution

 

[Steely Dan]

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

 

[Pyroadept]

Thanks!

 

[HallsofIvy]

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.

 

[Mentallic]

Homework Statement

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!

Homework Equations

The Attempt at a Solution

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.