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Why do you put that down as "disclaimer". This is why people make fun of us.

Why should 1/0 be +Infinity? Think of 1/x at the limit as x->0 from the left and from the right

I still think the 1/0 problem can be solved if we step outside Euclidian geometry. It is undefined for the very reason you give. The Numberphile YouTube channel has an excellent video [1] discussing why x/0 is undefined and worth watching.

[1] https://www.youtube.com/watch?v=BRRolKTlF6Q

With IEEE floats, 0 has a sign which can be used to denote the direction of the limit; if you try 1/(-0) in any program or language with sane numerics (i.e. not Excel or other dumb calculator emulators), it should give you -Infinity.

Bingo. If I'm in front of a computer, Mathematica. If not near one, then Wolfram|Alpha app/PCalc Lite on my iPad, else TI-83. But I've had my eyes on the HP Prime for a while. That'd be a huge jump from the 83!