The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in l 2 ( N ) given by some infinite symmetric Jacobi matrices. The approach used to calculate an asymptotic behaviour of eigenvalues is based on method of diagonalization, Janas and Naboko’s lemma [J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36(2) (2004) 643–658] and the Rozenbljum theorem [G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, (Russian) Trudy Maskov. Mat. Obshch. 36 (1978) 59–84]. The asymptotic formulas are given with use of eigenvalues and determinants of finite tridiagonal matrices.