# Undetermined vs. Undefined [duplicate]

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This often comes up in precalculus and calculus, that is sometimes an expression will be said to undefined while at other times undetermined. What is the fundamental difference between the two?

For example, division by zero $x/0$ is said to be undefined, while $0/0$ is said to be undetermined.

## marked as duplicate by Jyrki LahtonenAug 3 '15 at 11:18

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• I had a thought: Is it that you cannot extract more information from an expression that is in indeterminate form but can with an expression in undetermined form? – Red Aug 2 '15 at 21:34
• – Asaf Karagila Aug 2 '15 at 21:53
• I think there will come a time when we must decide to create a tag of $0/0$ questions or create a bot to delete them, because it is getting quite repetitive. – Aloizio Macedo Aug 3 '15 at 0:08

## 3 Answers

Division by zero is undefined in every case.

In calculus, the phrase “$0/0$ is an indeterminate form” means that you have a limit of the form $$\lim_{x\to a}\frac{f(x)}{g(x)}$$ where $$\lim_{x\to a}f(x)=0 \qquad\text{and}\qquad \lim_{x\to a}g(x)=0$$ but $f(x)/g(x)$ is defined in a set having $a$ as a limit point (usually, but not necessarily, a punctured neighborhood of $a$) and nowhere you do $0/0$, which makes no sense. In this case you can apply no standard theorem on limits and the limit, if existing, must be computed with some different technique than simply substituting the value $a$.

Some say that the value of the fraction $0/0$ (no reference to limits) is undetermined, but this has no real usefulness.

• $0^n$ for $n<0$ is undefined and $0^0$ is undetermined, right? – Money Oriented Programmer Jul 29 at 16:25
• @GoldDiggingProgrammer $0^0=1$, by definition of power with exponent $0$. In the context of limits, one has to remember that "a limit of the form $f(x)^{g(x)}$ with both function having limit $0$ has no general theorem available”, so it's common to say in the context of limits that $0^0$ is an indeterminate form. – egreg Jul 29 at 17:35
• Thank you very much! – Money Oriented Programmer Jul 30 at 5:50

$x/y=z$ means that $zy=x$, i.e. $x/y$ is the number you multiply by $y$ to get $x$. For $1/0$, there is NO number you can multiply by zero to get 1. For $0/0$, there is no UNIQUE number you can multiply by zero to get 0.

Without looking too deeply into it, I would first propose that undefined is more along the lines of not solvable. For ex: x/0 is not solvable until we know the definition of x. Undetermined is not yet solved. 0/0 is not yet solved, but we can use L'Hopital's rule to (often) eventually solve it.

(I'd like to caveat that this is an initial impression, so make of the answer what you will)