Abstract: For a different parameter $b$, let $l_G(b)$ denote the second largest root of $b\mu(\mu-2)\!-\!(\mu-1)^2(\mu-3)\!=\!0$ $(b\!=\!0,1,\cdots)$ and $l_T(b)$ denote the second largest root of $b\mu(\mu-2)\!-\!(\mu-1)^2(\mu-3)\!-\!(\mu-1)(\mu-2)\!=\!0$$(b\!=\!0,1,\cdots)$. Firstly, we will prove that there exist sequences of graphs  $\{G_{n,b}\}(b\!=\!0,1,\cdots)$ and $\{T_{n,b}\}(b\!=\!0,1,\cdots)$ such that their limit points of the third largest Laplacian eigenvalues are $l_G(b)$ and $l_T(b)$, respectively. Secondly, we will prove that $l_G(b)$, $l_T(b)$ and $2$ are all of the limit points of the third largest Laplacian eigenvalues which are no more than 2

Key words: Laplacian eigenvalue, characteristic polynomial, limit point