Nonhomogeneous Equations & Resonance · Hard problem

Every solution of $y''+5y'+6y=12+4e^{-t}$ has the form $y=c_1e^{-2t}+c_2e^{-3t}+2+2e^{-t}$ for constants $c_1,c_2$ fixed by the initial conditions. Find $\displaystyle\lim_{t\to\infty}\dfrac{y(t)-2}{e^{-t}}$ (the same for every choice of $c_1,c_2$).