[Math] Justifying why 0/0 is indeterminate and 1/0 is undefined


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

$\dfrac 10=x$
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 $-\infty$.

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