We study limit points of the spectral radii of $A_α$-matrices of graphs. Adapting a method used by J. B. Shearer in 1989, we prove a density property of $A_α$-limit points of caterpillars for $α$ close to zero. Precisely, we show that for $α\in [0, 1/2)$ there exists a positive number $τ_2(α)>2$ such that any value $λ> τ_2(α)$ is an $A_α$-limit point. We also determine the existence of other intervals for which all its points are $A_α$-limit points.