Is infinity / infinity equal to 1?

• MHB
In summary, Pradyumna's attempted proof of infinity / infinity = Gamma(1) = ProductPI(0) = 1 was not successful. He tried to formalize the notion of infinity as an "ordinary number" by introducing the "extended real number line" $\overline{\mathbb{R}}$ and topologizing it with the order topology. However, the algebraic properties of $\overline{\mathbb{R}}$ are not very good, and expressions such as $\frac{\infty}{\infty}$ are still undefined.
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?

I tried to prove that

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

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...

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.

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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?

I tried to prove that

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

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".

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.

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 "$\frac{\infty}{\infty}$".

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1. Is infinity divided by infinity equal to 1?

No, infinity divided by infinity is considered to be undefined. This is because infinity is not a specific number and can represent a limitless quantity, making it impossible to accurately divide or compare.

2. Why is infinity / infinity not equal to 1?

As mentioned, infinity is not a specific number and cannot be treated as such in mathematical operations. Dividing infinity by infinity would require defining both infinities as finite values, which is not possible.

3. Can infinity be equal to a finite number?

No, infinity is a concept that represents something without limits or boundaries. It cannot be expressed as a finite number, as it has no defined value. However, certain mathematical concepts and equations may involve infinity as a limit or asymptote.

4. Is infinity a real number?

No, infinity is not considered a real number. Real numbers are defined as any number that can be represented on a number line, and infinity cannot be accurately plotted on a number line.

5. What is the value of infinity?

Infinity is not a value that can be defined or measured. It is a concept that represents something without limits or bounds. Therefore, it does not have a specific numerical value.

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