The first question you need to ask is: What does "a/b " mean?

The answer is: "a/b is the bz=a ." (I'm using z as the unknown, since you are using x for other things).

Given that answer, let's discuss your points out of order:

(3) is perfectly fine:0/x , with x≠0 , is the solution to xz=0 ; the unique solution is z=0 , so 0/x=z . The reason it's x≠0 , so the only way for the product to be 0 is if z is 0 .

In (1), by "impossible" we mean that the equation that defines it has no solutions: for something to be equal tox/0 , with x≠0 , we would need 0z=x . But 0z=0 for any z , so there are x/0 ". So x/0 does

In (2), the situation is a bit trickier; in terms of the defining equation, the problem here is that the equation0z=0 has z as a solution (that's what the "infinite solutions" means). Since the expression a/b means "the bx=a , then when a=b=0 , you don't have a unique answer, so there is no "unique solution".

Generally speaking, we simply do not define "division by0 ". The issue is that, once you get to calculus, you are going to find situations where you have two a and b , and you are considering a/b ; and as a and b changes, you want to know what happens to a/b . In those situations, if a is approaching x and b is approaching y≠0 , then a/b will approach x/y , no problem. If a approaches x≠0 , and b approaches 0 , then a/b does not approach a and b approach 0 , then you don't know what happens to a/b ;
it can exist, not exist, or approach pretty much any number. We say
this kind of limit is "indeterminate". So there is a reason for
separating out cases (1) and (2): very soon you will see an important
qualitative difference between the first kind of "does not exist" and
the second kind.

The answer is: "

**unique**solution to the equationGiven that answer, let's discuss your points out of order:

(3) is perfectly fine:

*unique*is becauseIn (1), by "impossible" we mean that the equation that defines it has no solutions: for something to be equal to

*no*solutions to the equation. Since there are no solutions to the equation, there is no such thing as "*not*represent any number.In (2), the situation is a bit trickier; in terms of the defining equation, the problem here is that the equation

*any*value of**unique**solution toGenerally speaking, we simply do not define "division by

*variable*quantities,*anything*(the "limits does not exist"). But if*both*
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