= Solution
The <homogeneous linear differential equation> has roots $(-1\pm i\sqrt3)/2$, so its <complementary function> decays:
$$
f_h=e^{-t/2}\left(C_1\cos\frac{\sqrt3t}{2}+C_2\sin\frac{\sqrt3t}{2}\right).
$$
For a <particular integral>, set $f_p=B\cos2t+C\sin2t$. Substitution gives
$$
f_p''+f_p'+f_p=(-3B+2C)\cos2t+(-2B-3C)\sin2t.
$$
The two coefficient equations are $-3B+2C=A$ and $-2B-3C=0$, so $B=-3A/13$ and $C=2A/13$. Consequently every solution obeys
$$
\boxed{f(t)=\frac A{13}(-3\cos2t+2\sin2t)+O(e^{-t/2}),\qquad t\to\infty.}
$$
The limiting behaviour is a periodic steady oscillation with amplitude $|A|/\sqrt{13}$, the long-time response of a <forced harmonic oscillator>. Unless $A=0$, there is no finite numerical limit: at $t=n\pi$ and $t=n\pi+\pi/2$, the two limiting values are $-3A/13$ and $3A/13$. For $A=0$, $f(t)\to0$.
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