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.
On
$x/y=z$ 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.
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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)
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.