I am a little bit surprised to see the confusion between infinity, undefined here taking 0/0 etc.

 

Infinity is an undefined quantity and thus lies outside the set of Real Numbers and thus can't be defined quantitatively despite the fact that it has a qualitative meaning.

 

The 7 forms 0/0, etc. are "indeterminate" forms. They can't be determined and so the question of definition does not arise.

 

A "divide by zero" is essentially undefined and not strictly infinity because infinity itself is undefined. You can't give an answer and say "this is it" when the answer itself is not defined. That is why limits which evaluate to "infinity" are essentially answered as "limit does not exist" and not "limit is infinity" because limit gives us a quantitative idea of the function and infinity is not quantitatively defined.