I'm given a physical model of something. Say it's a radius of a planet:
$$R=\pi\left(\frac{n}{2\pi G}\right)^{1/2}$$ where $G$ is the universal gravitational constant, n - "positive constant"
If I don't have any information about n aside from the fact that it's positive, how can I make a conclusion of finiteness of R? if n goes to infinity, R would go to infinity.. am I missing something?
All of the things on the right hand side are finite; therefore, the left-hand side is finite.
Your confusion may stem from the similar, but distinct, notions of "arbitrarily large" and "infinite". It is true that we may set $n$ to be arbitrarily large (but finite!); however, this is different than assuming it to be infinite.
When we assume something to be arbitrarily large, we say that the relationship is true/must hold for any finite value, no matter how large we assume it to be. If we want to choose a bigger quantity, that's fine. But that quantity, too, is finite. In other words, no matter how large some finite $n$ is, $n+1$ is always bigger. This is not true for $n$ being infinite.