Order of evaluating an a fraction's components - 0/0

[Square1]

Just wondering if I've forgotten a rule here, or there is some new terminology I can learn.

We know 0/x when x > 0, is equal to 0. x/0 is undefined..since we "blow up" dividing any value by a value that is more than infinitely small...by zero.

We say that 0/0 is also undefined. We choose to consider the denominator 0 here first to say, "dividing by zero...can't be defined", instead of first considering the numerator and saying maybe, "zero is going to be divided. It's going to be equal to zero no matter what since we started with nothing".

Question: Is there an algebraic rule, or convention, that generally states you should start to evaluate the denominators first? Or is x/0 simply its own case where we can begin and end evaluating the parts that make up an expression?

 

[Mark44]

Just wondering if I've forgotten a rule here, or there is some new terminology I can learn.

We know 0/x when x > 0, is equal to 0. x/0 is undefined..since we "blow up" dividing any value by a value that is more than infinitely small...by zero.

The basic rule is that division by 0 is not allowed.

We say that 0/0 is also undefined.

0/0 is called an indeterminate form. It shows up in limits where both the numerator and denominator are approaching zero. This is indeterminate, because some quotients with this form actually have a limit, which can be literally any number or even $\infty$ or $-\infty$.

We choose to consider the denominator 0 here first to say, "dividing by zero...can't be defined", instead of first considering the numerator and saying maybe, "zero is going to be divided. It's going to be equal to zero no matter what since we started with nothing".

Question: Is there an algebraic rule, or convention, that generally states you should start to evaluate the denominators first?

No. The rule is that division by zero is not allowed.

Or is x/0 simply its own case where we can begin and end evaluating the parts that make up an expression?