This paper investigates the application of approximation schemes for differen-tial-difference equations [1-3] to construct algorithms for the approximate fin-ding of nonsymptotic roots of quasipolynomials and their application to study the stability of solutions of systems of linear differential equations with delay. Consider the initial problem for a system of differential-difference equations dx dt = Ax(t) + k i=1 Bix(t − τi), (1) x(t) = φ(t), t ∈ [−τ, 0], (2) where A,Bi, i = 1,k fixed n × n matrix, x ∈ Rn, 0 < τ1 < τ2 < ... < τk = τ. Let us correspond to the initial problem (1) - (2) the system of ordinary differential equations [1-2] dz0(t) dt = A(t)z0(t) + k i=1 Bizli (t), li = [ τim τ ], dzj(t) dt = μ(zj−1(t) − zj(t)), j= 1,m, μ = m τ ,m ∈ N, (3) zj(0) = φ(− τj m ), j= 0,m. (4) Theorem 1 [2]. Solution of the Cauchy problem (3)-(4) approximates the solution of the initial problem (1)-(2) at t ∈ [0,T] if m→∞. Theorem 2 [1]. If the zero solution of the system with delay (1) is exponentially stable (not stable), then there is m0 > 0 such that for all m> m0, the zero solution of the approximating system (3) is also exponentially stable (not stable). If for all m> m0 the zero solution of the approximation system (3) is exponentially stable (not stable) then the zero solution of the system with a delay (1) is exponentially stable (not stable).It follows from Theorem 2 that the asymptotic stability of the solutions of the delayed linear equations approximating system of ordinary differential equations for sufficiently large values of m are equivalent. This fact will be used to study the stability of linear differential-difference equations [3].