It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by $\infty$. The resulting compactification is homeomorphic to a circle in the plane.
Let $x\neq0$ be a "finite" real number. Are the expressions $x\cdot\infty$ and $x/0$ are well defined in $\mathbb R\cup\{\infty\}$? In that case, can we say that $x/0=\infty$ and $x\cdot\infty=\infty$ ?
One has to pay attention when formulating a question like this: The one-point compactification is a purely topological procedure: $\mathbb{R} \cup \{\infty \}$ is a topological space and it does not come with additional structure. You're asking if one can endow the space with additional structure that is more-or-less sensible (whatever that means; but it particular it should extend the usual structure on the field $\mathbb{R}$).
Indeed, there are way of doing so and here is one typical such extension that I just copied from Wikipedia since it is too long. It plays a relevant role in measure theory.
In particular, your hopes $x/0 = \infty$ and $x \cdot \infty$ are answered affirmatively in case $x \neq 0$. The expressions $0/0$ and $0 \cdot \infty$ remain undefined.