Why is divide by zero not the square root of infinity?

Question

Searching through past posts I didn’t see anything specific to this. As I understand it, we can approximate what occurs when dividing by zero if you graph the function [y=1/x](https://i.imgur.com/MebU9l3.png) (ripped from google)

As X approaches zero, it becomes both infinity or negative infinity, which results in it being undefined.

Couldn’t positive or negative infinity be defined as the square root of infinity? or the square root of infinity squared? Obviously not all infinities are equivalent.

Thanks

Sincerely, a person who failed math

In: 0

Answer 1

It’s very important in math to say exactly what we mean in order to avoid confusion about things like this question!

Say what you’re saying, but be more precise about it. As x approaches zero from the left, y approaches (not becomes) negative infinity. As x approaches zero from the right, y approaches (not becomes) positive infinity. There is nothing wrong with this happening. It does mean that the limit as x approaches 0 is undefined, which again is perfectly acceptable as a result. Nothing has to happen in math just because it feels like it should. Instead, what happens, happens.

Note that you are using the word “approaches” very loosely, but it has a very technical definition. When we say something approaches infinity or negative infinity we in no way treat infinity as a number, or define it at all even. It’s typical to gloss over these technicalities in a first year calculus class.

Creating a new definition is very tricky. If you want to define positive or negative infinity as the square root of infinity then you need to be clear about what this even means. You can define anything to be anything, but this can lead to meaningless results.

If you try to define a thingy called “inf” such that sqrt(inf) = inf = -inf then you can see how it plays with numbers. You will probably find that it creates paradoxical situations, which means that the definition does not play well with the rules that numbers follow. You may need to get rid of other important rules in order for “inf” to be treated as a number.