There are also a couple patterns that help conclude that 1/0 should be regular infinity and negative infinity.
Namely, the graph for 1/x. Here, as x approaches 0, f(x) approaches both infinity and -infinity, depending on which side you look at.
Furthermore, defining division by 0 as any number yields 1 = 2:
1/0 = absolute infinity = 2/0
1/0 = 2/0 | cancel out the 0
1 = 2
Which is fine since infinity isn't a number.
But it not being a number also means that it is rather nonsensical to represent it in the graph for f(x) since we now have a graph that, for the most part, consists of numbers but then at one point, decides to show a cardinality as a vertical line, which would usually read as "every number at once".
Maybe that is a valid interpretation of absolute infinity but it still means that we're now plotting two incomparable concepts in the same graph for no good reason.