Why is $0/0$ not $\Bbb R$? Whenever I asked my grade school and college teachers why $\frac00$ is undefined, they would show me a graph of $\frac1x$ and point out the vertical asymptote at $0$, noting that "Since it approaches $-\infty$ from the left and $\infty$ on the right, it could be $-\infty$, $\infty$, or anything in between. So, we say it's undefined."
My question is, then, why isn't it $\Bbb R$, or even $\{\Bbb{R,C}\}$?

For the same reason that $\lim_{x\to \infty} \sin(x)$  is not defined, rather than $[-1,1]$.
The Epsilon-Delta definition of the limit requires that the limit is an element of the set, and not a set itself.
