We study limit points of the spectral radii of Laplacian matrices of graphs. We adapted the method used by J. B. Shearer in 1989, devised to prove the density of adjacency limit points of caterpillars, to Laplacian limit points. We show that this fails, in the sense that there is an interval for which the method produces no limit points. Then we generalize the method to Laplacian limit points of linear trees and prove that it generates a larger set of limit points. The results of this manuscript may provide important tools for proving the density of Laplacian limit points in 4.38+,∞). •We investigate the Hoffman concept of limit points for the Laplacian matrix of a graph. The problem is to determine which real numbers are limit points of the spectral radius of these matrices.•We make progress towards the proof that any real number larger than 4.38+ is such a limit point.•We adapt Shearer's method, used to solve the problem for the adjacency matrix, extending it to linear trees.•We determine larger sets of limit points and provide technical tools to prove the density of Laplacian limit points in 4.38+,∞).