Suppose we have two sequences of rational numbers, $(p_i)_{i=1}^\infty$ and $(q_i)_{i=1}^\infty$, and suppose $$\lim_{i\to\infty}\frac{p_i}{q_i}=c<\infty,$$ where $c$ is known. Are there any sufficient conditions of the two sequences of rationals to say anything about $c$, in particular whether or not $c$ is rational, irrational, or transcendental?
EDIT: I forget to say that $p_i,q_i\to\infty$ as $i\to\infty$.
It is useless to attempt to determine the nature of such a limit simply from the nature of the respective sequences.
If both $p_i$ and $q_i$ converge, then $\lim_{i\to\infty}\frac {p_i}{q_i} = \frac{\lim p_i}{\lim q_i}$ is simply the quotient of two real numbers, because any convergent sequence of rational numbers (ie any Cauchy sequence) uniquely defines some real number. Determining the integrity, the rationality or the algebraicity of the quotient is no different than of any quotient of two real numbers.If one of $p_i$ and $q_i$ diverges and the other converges, then clearly $p_i$ must be the convergent sequence (otherwise the limit of the quotient diverges), and thus the limit is $0$.
If both $p_i$ and $q_i$ diverge, then for $\frac{p_i}{q_i}$ to converge we must have that $p_i \propto q_i$, but the type of proportionality constant is not easy to determine.
For example, take $p_i = 6^{i}$ and $q_i = \left\lfloor\frac{6^{i}}{r}\right\rfloor$ for some $r \in \mathbb R$, then $\lim_{i\to\infty}\frac{p_i}{q_i} = r$, and even though both sequences are integer sequences of exponential growth, the limit of their quotient can be any real number.