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.
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.
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?
Thanks in advance!