Let $\{a_n\},\{b_n\},\{c_n\}$ be sequences of positive real numbers such that $\lim_{n\to\infty}c_n=0$. Is it the case that the sequences
$$ \left\{\frac{a_n-b_n}{a_nc_n}\right\} \qquad\text{ and }\qquad \left\{\frac{a_n-b_n}{b_nc_n}\right\} $$ have the same set of limit points in the extended real line $\mathbb R\cup\{-\infty\}\cup\{\infty\}$?
Yes.
Suppose $\frac{a_{n_k}-b_{n_k}}{a_{n_k}c_{n_k}} \to +\infty$. Then, $b_{n_k} < a_{n_k}$ for large $k$, so $\frac{a_{n_k}-b_{n_k}}{b_{n_k}c_{n_k}} \ge \frac{a_{n_k}-b_{n_k}}{a_{n_k}c_{n_k}} \to +\infty$. So, $\frac{a_{n_k}-b_{n_k}}{b_{n_k}c_{n_k}} \to +\infty$. Similarly with $-\infty$.
Now suppose $\frac{a_{n_k}-b_{n_k}}{a_{n_k}c_{n_k}} \to \lambda \in \mathbb{R}$. Then, since $\frac{1}{c_{n_k}} \to +\infty$, it must be that $\frac{a_{n_k}-b_{n_k}}{a_{n_k}} \to 0$, i.e., that $\frac{b_{n_k}}{a_{n_k}} \to 1$. Therefore, $\frac{a_{n_k}-b_{n_k}}{b_{n_k}c_{n_k}} = \frac{a_{n_k}}{b_{n_k}} \frac{a_{n_k}-b_{n_k}}{a_{n_k}c_{n_k}} \to \lambda$.