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Is (2/infinity = 1/infinity)?
Essentially, you gave the answer yourself: "infinity over infinity" is not defined just because it should be the result of limiting processes of different nature. I.e., since such a definition would be given for the sake of completeness and coherence with the fact "the limiting ratio is the ratio of the limits", your
and, say (this is my choice)
would have to be equal (as they commonly define 
), which does not happen.
I will quote the following from Prime obsession by John Derbyshire, to answer your question.
Nonmathematical people sometimes ask me, “You know math, huh? Tell me something I’ve always wondered, What is infinity divided by infinity?” I can only reply, “The words you just uttered do not make sense. That was not a mathematical sentence. You spoke of ‘infinity’ as if it were a number. It’s not. You may as well ask, ‘What is truth divided by beauty?’ I have no clue. I only know how to divide numbers. ‘Infinity,’ ‘truth,’ ‘beauty’—those are not numbers.”
I object to your teacher's answer (even if we remove physical obstructions, such as indivisibility of some particle or another): after any finite amount of time, Aladdin will have only divided it into pieces of size $(1/2)^n$ for some finite $n$; this is still positive. However, the limit is zero, which is what is meant by $(1/2)^\infty$. As $n$ gets bigger, $(1/2)^n$ gets as small as you want, so we say that its limit as $n \rightarrow \infty$ is zero.
What your teacher said was not very mathematically exact. You have to observe the infinite sequence of powers of one half,
$$1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\cdots$$
And the proper term is that the sequence tends to zero (it converges to its limit which is $0$).
The fact that this goes to zero shouldn't surprise you. But you need an infinite number of cuts.
Of course this is assuming that he's throwing the rest of the carpet away. Otherwise you just get a heap of differently sized pieces :)
Okay a little background first, I'm a master's level Psychology student and have done very poorly in math since school but I've recently realised I enjoy Stats and am actually good at it, so I'm giving math a second shot and trying to learn it online.
While learning, this simple series came to my head that as you divide a number with another same number but added with 1 (n/n+1), it gets closer and closer to .99.
For example, 1/2 = .50, 3/4 = .75, 8/9 = .88, 9/10 = 0.9 and so on. I then carried out the same curiousity for larger numbers, 12000 to be exact, and the numbers kept approaching 1 (I don't have much knowledge of limits but those who do might relate this to it).
That's when it struck me, if we keep going infinitely, the resulting number has to be 0.9999 (recurring infinitely). There are several different proofs for why 0.9 recurring = 1 so I'm not going to add those here, but this is an interesting thought I had but since my knowledge of math is very limited, I want to know if my point is valid or if I've made some error in my reasoning.
Edit: Thanks for all the helpful comments, and for all those who took this opportunity to be condescending, congrats you're better at math than someone who knows very little about it (how about a race with a turtle next?). Also, it's been pointed out to me that I miswrote the equation, it ought to be ∞/(∞+1) and apparently that makes a big difference, for reasons I hope to understand. Hope it brings a little more clarity.