Infinity In Math - The Nick Lim Proposal

Question

So infinity is clearly a very strange, concept.

So I have the following proposal (the nick lim proposal) which can not be solved ( at least to my current knowledge, hopefully you can shed some light ).

For the following scenarios since I do not have an infinity character on my keyboard, replace x with the infinity symbol.

What is x/x?

Well we know 2x = x

So we could rewrite x/x = 2x/x

Well 2x/x

Well in that scenario x/x = x

But

lets flip that around

2x = x so x/x = x/2x

In this scenario x/x = 1/2 //Think of limits imagine y/2y as y approaches infinity it's 1/2.

So it seems x/x = "1/k where k is all positive integers" or "just x"

Edit: 1.1x = x and 1.5x = x and .1x = x

So x/x = "1/k where k is all positive real numbers" or "just x"

What do you think?

Answer 1

What would you think of the reasonning " $0 \times 1 = 0 \times 2$ so $1 = 2$" ? Both products $= 0$, so you can't simplify.

Similarily $\infty \times 1 = \infty \times 2$ cannot implies 1 = 2 for the same reason: both products are infinity, so you can't simplify.

Answer 2

$\infty$ is the limit of t at infinity. $\infty$ is also the limit of 2t at infinity.

When you write $\frac \infty \infty$, there is no reason that both $\infty$ are set to the same sequence, each could freely map to any sequence going to infinity. That's why the result is indeterminated, and that's why your reasonning is flawed.