[belter]

on 1.2) is 1 divided by zero not undefined instead of infinite?

[kzrdude]

Depends on "which numbers you use". See for example https://en.wikipedia.org/wiki/Riemann_sphere where the extended complex numbers are used.

[prvc]

It is convenient to posit, or "define" a (unique) point at infinity which is the inverse of 0. That way, a lot of propositions work out without extra special cases. It was misleading of the author to just say "pretend infinity is a real number".

[renewiltord]

It’s just some sleight of hand. You can switch all of the parts to be lim x to zero of 1/x m.m. if you wish to satisfy yourself