What is infinity to the zeroth power?

Question

I am not happy with the answers posted to similar questions.

For example, in:

What is infinity to the power zero

the accepted answer is 1, which is definitely wrong.

I think the answer is any non-zero, non-one, non-infinite number. Is this correct?

Looking for the solution to: $(1/0)^0$

Please either remove the downvotes or delete the answer. (One or the other, I don't care.....) — Jiminion, Apr 28 '16 at 16:00
(For future reference) you are expected to provide context. You stated that you are unhappy with answers posted to similar questions. You should *reference* those posts and specify which parts are unsatisfactory to you. — Em., Apr 28 '16 at 16:05
@Jiminion I think neither will happen, in the short term. As for the reason I think this question is unclear: I don't see why you think what you suggest is true, since actually $\lim_{k\to\infty} k^{1/k}=1$. — , Apr 28 '16 at 18:35
The question didn't ask about the limit. $1^\infty = 1$. The limit is problematic because infinity is not a number. — Jiminion, Apr 28 '16 at 19:25

SchrodingersCat · Accepted Answer · 2016-04-28T15:53:41.177

4

It is an indeterminate form and as such cannot be assigned any value.

It is better expressed as $\lim_\limits{{x\to \infty}\\,\\{y\to 0}}x^y$.

As commented by Did, it is true that $x^y$ has no limit when $x\to \infty$ and $y\to 0$.

And $\infty^0$ has no definite meaning in mathematics. It is basically some kind of a meaningless statement where the notation of infinity has been wrongly used since infinity is not a number, it is a concept. To speak of infinity, I must add the following:

In standard real analysis, the symbol $\infty$ is simply used to denote an unbounded limit. Whenever the symbol is used, in series and integrals, for example, it has a precise definition with epsilon/delta. Similarly, in set theory, it also has a precise definition; we say a set $S$ is infinite if there is no bijection from $S$ to a bounded subset of $\mathbb{N}$.

edited Apr 28 '16 at 15:53

answered Apr 28 '16 at 15:11

SchrodingersCat

24,472

Reason for Downvote??? – SchrodingersCat Apr 28 '16 at 15:12
After the (good) initial sentence, I was expecting to see the argument that $x^y$ has no limit when $x\to\infty$ and $y\to0$, unfortunately, next, we are summoned to believe that the "better" explanation of the limit is as the limit of $x^{1/x}$ (an assertion with which one can disagree). – Did Apr 28 '16 at 15:24
2

As asked, what the question poses isn't even, in my opinion, an indeterminate form as usually understood in limits. Perhaps this answer can mislead someone in believing $;\infty^0;$ has some definite meaning in mathematics, which I think it doesn't. – DonAntonio Apr 28 '16 at 15:25
1

@Joanpemo: I agree. I was thinking of writing something similar until I saw your comment. – Dave L. Renfro Apr 28 '16 at 15:28
@Joanpemo Is it better addressed now? – SchrodingersCat Apr 28 '16 at 15:54
@Did Do you want me to add some more information? – SchrodingersCat Apr 28 '16 at 15:54
I would say the precise definition of $inf$ would be $1/0$. – Jiminion Apr 28 '16 at 15:55
1

@Jiminion No, not at all. Infinity is a mathematical concept. It seriously does not mean $\frac{1}{0}$. – SchrodingersCat Apr 28 '16 at 15:57
The current version is much better. One remaining problem is that you use "concept" in a bizarre way (and one which would make every philosopher cringe...). – Did Apr 28 '16 at 15:58
It can mean more than one thing. That is one of them. (Note that I agree that it is not a number.) – Jiminion Apr 28 '16 at 15:58
@Did Why would you say that? I did not want to mean something bizarre. – SchrodingersCat Apr 28 '16 at 15:59
@Jiminion Note that division by zero is not defined and hence $\frac{1}{0}$ is meaningless and undefined. Do you get it now? – SchrodingersCat Apr 28 '16 at 16:00
@SchrodingersCat I think it is better now thought the very first words "it is an..." screech to me. – DonAntonio Apr 28 '16 at 16:01
We'll have to agree to disagree. Using limits, $1/0$ is pretty clearly either infinity or negative infinity. – Jiminion Apr 28 '16 at 16:02
1

@Jiminion Your use of the word "or" already makes what you say, in my opinion, pretty non-clear – DonAntonio Apr 28 '16 at 16:03
$1/0$ is pretty clearly either infinity and/or negative infinity. Either or both. – Jiminion Apr 28 '16 at 16:16

What is infinity to the zeroth power?

1 Answers1