Interpretations of $\frac{\infty}{\infty}$

Question

I am trying to understand the physical sense of the mathematical construct $\frac{\infty}{\infty}$

Suppose we have a function $f(x)$ representing some physical construct depending on a "quantity" $x$ and a function be $g(x) = \frac{f(x)}{f(x)}$

Now I understand mathematically it is not possible to define $g(x)$ when $f(x) = 0$ or $f(x) = \infty$ as they are deemed indefinite. But shouldn't there definition be subject to the definition of the construct $\infty$ ?

in physics various times we define infinite length as simply a very big length compared to a very small length (say infinite solenoid for calculation of field at center of a very long solenoid) in such cases inifinity as I see, seems to be just a big number which would simply give $\frac{\infty}{\infty}$ as $1$

Is my line of thought correct ? Is it true that for a physical manifestation of constructs such as $\frac{\infty}{\infty}$ we need to define the physical nature $\infty$ ?

What would be the result when we define $\infty$ as the distance at which observable fields of particles (say gravitational field of point mass m) becomes $0$ ?

If there is actually a difference in the mathematical and physical manifestations, then why is that ?

Answer 1

Basically, in mathematics, "$\infty$" is just a symbol. It is not a number nor an object, and we cannot use it in formulas, meaning you cannot divide with it or add and subtract it. As a symbol, it is used if a value exceeds all bounds, for example the limit of $1/n$ as $n$ goes "to infinity" is zero.

Another way it is used is when a function (or, similarly, a sequence) $f(x)$ has the property that for any upper bound $M$, the value $f(x)$ is greater than $M$ for $all$ values $x$ from some point on. In that case, we write $\lim_{x\rightarrow \infty}f(x) = \infty$. Be careful, as this "equals" sign does NOT mean the equality of two real numbers. As standard limits go, the limit of $f(x)$ does NOT exist and is definitely NOT a real number.

It's even more risky to try and perform calculations with $\infty$. For example, you say that you feel like $\frac{\infty}{\infty}$ should be $1$, and in fact, if you take $f(x)=g(x)$ such that $\lim_{x\rightarrow\infty}f(x) = \infty$, that's what you get if you calculate the limit of $f(x)/g(x)$. Howerver, take $f(x) = x$ and $g(x) = 2x$. Now, with some dodgy algebra, you have

$$ 1 = \frac{\infty}{\infty} = \frac{\lim_{x\rightarrow\infty}f(x)}{\lim_{x\rightarrow\infty}g(x)}=\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\lim_{x\rightarrow\infty}\frac{x}{2x}=\frac{1}{2} $$ which you probably agree is not right.

Answer 2

Symbols like $\frac{\infty}{\infty}$ and $\infty - \infty$ are often used informally to classify "types" of expressions in a limit.
However, they don't reveal anything about the value of the limit.

For example: $$\lim_{x \rightarrow \infty} \frac{2x}{x}=2$$ $$\lim_{x \rightarrow \infty} \frac{x^2}{x}=\infty$$ And notice that both are "classified" as $\frac{\infty}{\infty}$.

Regarding the symbol $\infty$ itself appearing in relations (Real numbers for simplicity):
$\cdot =\infty$ is just a shorthand notation for "$\cdot$ is greater than any real number".
$\cdot <\infty$ is just a shorthand notation for "$\cdot$ is some real number".

Answer 3

Since every strictly positive number multiplied with infinity is infinity, and infinity times infinity is also infinity, it follows that $\dfrac\infty\infty$ can be any strictly positive value on the extended real axis $\overline{\mathbb{R}}$, which is why the operation is undefined. The same reasoning applies to $\infty-\infty$.