Let us consider a positive and bounded real sequence $(u_{n})$. Let $a,b$ be two positive real numbers. Assuming that there exists $N$ such for all $n>N$ we have $u_{n}>0$, define the new sequence:
$$v_{n}=\log(a+bu_{n})/u_{n}$$ for all $n>N$.
I am asking about the necessary steps to calculate limit inferior and limit superior of $(v_{n})$.
Well, I don't think theres an explicit way given what we know. If $L$ is a partial limit of $(u_n)$ achieved by a subsequence $(u_{n_k})$, then we can create a subsequence $(v_{n_k}=\frac{\log(a+bu_{n_k})}{u_{n_k}})$ of $(v_n)$ which approaches to $f(L):=\frac{\log(a+bL)}{L}$ by arithmetics. since $f$ is always monotonic on $(0,\infty)$, the corrospondence is bijective, meaning two partial limits of $(u_n)$ can't mix, and that any partial limit of $f(u_n)$ must come from a unique partial limit of $(u_n)$.
If so we can charecterize the partial limits of $f(u_n)$ by $$\left \{ f(L) | L\ \text{is a partial limit of}\ (u_n) \right \}$$ and we need to find its $\inf,\sup$. Note that this includes the special case of a partial limit $0$ which yields a partial limit of $-\infty/\infty/1$ depending on $a,b$.