Is infinity / infinity equal to 1?

[pradyumna1974]

Dear Members,
I tried to prove this indeterminate form of infinity / infinity as 1. I could come up a reasonable approach
with Gamma and Product functions. I posted my proof as video in Youtube. Here is the URL for the video.
I would like to receive feedback and challenges on where my approach is going wrong in this proof?

Youtube URL : https://www.youtube.com/watch?v=9W2LclHwQzs

I tried to prove that

infinity / infinity = (infinity ^ 0) = Gamma(1) = ProductPI(0) = 1

Thank you for your help!
Pradyumna

 

[Theia]

Just a comment, but how your attempt deals with following (\(\displaystyle a > 0\)):

\(\displaystyle \lim_{x \to \infty} \frac{ax}{x} = a \lim_{x \to \infty} \frac{x}{x} = a \cdot 1,\)

and one obtains that \(\displaystyle \frac{\infty}{\infty}\) can take all positive real values...

 

[Dethrone]

You can't do $\frac{\infty}{\infty}=\infty^{1-1}=\infty^0$ ... that is, treating infinity as if it were a number; otherwise, you'd get all sorts of weird results.

 

[tkhunny]

Dear Members,
I tried to prove this indeterminate form of infinity / infinity as 1. I could come up a reasonable approach
with Gamma and Product functions. I posted my proof as video in Youtube. Here is the URL for the video.
I would like to receive feedback and challenges on where my approach is going wrong in this proof?

Youtube URL : https://www.youtube.com/watch?v=9W2LclHwQzs

I tried to prove that

infinity / infinity = (infinity ^ 0) = Gamma(1) = ProductPI(0) = 1

Thank you for your help!
Pradyumna

Keep in mind that "infinity / infinity" is a FORM, not a NUMBER. Don't treat it like a number. "infinity / infinity" is a theoretical condition that never is actually attained. There is a very big difference between "increases without bound" and "we arrived somewhere finite".

 

[S.G. Janssens]

It is possible to formalize the notion of $\infty$ as an "ordinary number" by introducing the "extended real number line" $\overline{\mathbb{R}}$ and topologizing it with the order topology. In my limited experience, this is more of a convenience in certain areas such as real and convex analysis, and not so much a source of insight. Also, the algebraic properties of $\overline{\mathbb{R}}$ are not very good. In particular, expressions such as $\frac{\infty}{\infty}$ are still undefined, for reasons such as those pointed out nicely in post #2 and the other posts above.

 

[HOI]

You use the phrase "indeterminant form". Surely you understand that the reason that $\frac{\infty}{\infty}$ is called an "indeterminant form" is that taking the limit as the numerator and denominators go to infinity, in different ways will give different results. Certainly $\lim_{x\to\infty} \frac{x}{x}= 1$ but $\lim_{x\to\infty} \frac{ax}{x}= a$ for any value of a. all of which can be thought of as "$\frac{\infty}{\infty}$".