$\dfrac 00=x$

$0x=0$

$x$ can be any value, therefore $\dfrac 00$ can be any value, and is indeterminate.

$\dfrac 10=x$

$0x=1$

There is no such $x$ that satisfies the above, therefore $\dfrac 10$ is undefined.

**Is this a reasonable or naive thought process?**

It seems too simple to be true.

## Best Answer

Those expressions are about limits, not about numbers.

We say that $\frac00$ is an indeterminate form because a limit of that form can take any value:$$\lim_{y\to0}\frac{xy}y=x,$$for any real number $x$.

On the other hand, a limit of the type $\frac10$ cannot take any value. If it exists, it can only be $\infty$ or $-\infty$.