# Difference quotients with same limit points

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 $$\mathbb R\cup\{-\infty\}\cup\{\infty\}$$?