Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. Let $p_n$ and $q_n$ be the coprime natural numbers such \begin{equation*} \frac{p_n}{q_n}=\frac{a_n^{(n)}}{b_n^{(n)}}\left[=\binom{a_n+n-1}n\Big/\binom{b_n+n-1}n\right], \end{equation*} where $x^{(n)}:=x(x+1)\dots(x+n-1)$ is the Pochhammer rising factorial.

It then appears that \begin{equation} \tfrac1n\,\ln p_n\to f(a,b)\tag{1} \end{equation} for some positive real function $f$ and any distinct positive real numbers $a$ and $b$. (Here, in view of an immediate cancellation, such as $\frac{2\cdot3\cdot4}{3\cdot4\cdot5}=\frac25$, without loss of generality $a+1\le b$.)
This may be not hard to prove, using the prime number theorem (or maybe some refinement of it) together maybe with summation by parts.

Some easy remarks added: By Stirling's formula, $\tfrac1n\,\ln a_n^{(n)}\sim\ln n\to\infty$. However, because of the cancellations, $\frac1n\,\ln p_n$ seems likely to have a finite limit. Anyway, $\limsup_n \frac1n\,\ln p_n\le(a+1)\ln 2<\infty$, since $p_n\le\binom{a_n+n-1}n\le2^{a_n+n-1}=2^{(a+1+o(1))n}$.

However, $(1)$ may be well known. In such a case, it would be good to have a reference. Otherwise, it would be good to have a hopefully short and efficient proof, preferably with an explicit expression for $f(a,b)$. The case when $a$ and $b$ are natural numbers would be enough for my current needs (which arose in some work in approximation theory/numerical analysis).