Does division by zero imply a different type of division in this situation

Question

Firstly, I would like to apologize if this is somehow addressed in one of the many many explanations about why division by 0 is impossible that appear on this site. I have not yet found one that explains this type of situation, at least explicitly.

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Let's say that you're trying to ride a roller coaster as many times as you can. You have X dollars, and it costs Y dollars for each ride. In general, you can find the number of times you can ride by X/Y. However, if the cost of the roller coaster is free, Y=0, and you would create a divide by zero situation if you attempted to apply the same formula.

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Clearly X/0 is undefined, but for f(X,Y), f(X,0) will give infinity, instead of the undefined X/0. It is intuitive that when Y is 0, not only approaching 0, you can go on an unbounded number of rides, but how would this be mathematically shown if you can't use division to show it and a limit doesn't show that f(X,0) is not undefined?

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Thanks in advance!

Answer 1

In general, you can find the number of times you can ride by $X/Y$.

Right there is your mistake; that sentence should read "When $Y$ is nonzero, you can find the number of times you can ride by X/Y; when $Y$ is zero, the number of times you can ride is infinite." (or better, "is unbounded.")

Answer 2

If we "define" $$\frac{1}{0}=\infty$$ , we also have to "define "$$\frac{2}{0}=\infty$$ With the usual properties in the real numbers we would get $$1=0\cdot \infty=2$$ which is clearly a contradiction. To avoid this, the only possibility is to forbid division by $0$.