I am interested in any references and/or information regarding this sort of asymptotic relationship.
Given the composition:
$$\frac{f(x)h(x)+g(x)}{f(x)}$$
With the following polynomial functions $\ g(x)=\sum_{i=0}^ka_ix^i$ , $\ f(x)=\sum_{i=0}^nb_ix^i$ and an arbitrary function $h(x)$.
Assuming that $\lim_{x\to \infty}h(x)$ exists and $\text{deg(f)}$ $\ge$ $\text{deg(g)}$ than.
$$\frac{f(x)h(x)+g(x)}{f(x)} \sim h(x)+\frac{a_k}{b_n}$$
Noting that $\frac{a_k}{b_n}=0$ when $\text{deg(f)}$ $\gt$ $\text{deg(g)}$.