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the result is well defined in terms of a limit: lim 1/x as x -> 0 = ∞. I consider 1/0 = ∞ a practical shortcut notation.

Except it’s also negative infinity.

Oh, right!

lim (x->0⁺) 1/x = ∞

Actually, ∞. +∞.and -∞ are three different things.

∞ and +∞ are the same thing afaik.

It depends on the context. ∞ is sometimes defined as the point on the projective real line equal to both +∞ and -∞.

Technically ∞ is an element of the projective real line: R U {∞}. +∞ and -∞ are distinct elements in the extended real number system: R U {-∞, +∞}.