= Solution
Write $A=aI+B$, where $B=\bigl(\begin{smallmatrix}1&-2\\1&-1\end{smallmatrix}\bigr)$ has $B^2=-I$. One eigenpair is
$$
\boxed{\ell=a+i,\qquad e=\begin{pmatrix}1+i\\1\end{pmatrix}.}
$$
Because $A$ is real, conjugating $Ae=\ell e$ gives $A\bar e=\bar\ell\bar e$. The <eigenvector> matrix $(e,\bar e)$ has <determinant> $2i\ne0$, so the <eigenvectors> form a <basis>. Reality of $z$ then forces conjugate coefficients, giving $z=\alpha e+\bar\alpha\bar e$.
Set $c=(1-i)/2$. The forcing has the same basis decomposition,
$$
h(t)=c e^{-it}e+\bar c e^{it}\bar e.
$$
Equating the coefficient of $e$ in the <linear system of differential equations> yields
$$
\boxed{\dot\alpha+(a+i)\alpha=c e^{-it}.}
$$
For $a>0$, an <integrating factor> gives $\alpha=(c/a)e^{-it}+C e^{-(a+i)t}$. The zero initial vector means $\alpha(0)=0$, so
$$
\alpha(t)=\frac c a(1-e^{-at})e^{-it}.
$$
Taking the real vector combination gives
$$
\boxed{z(t)=\frac{1-e^{-at}}a
\begin{pmatrix}2\cos t\\\cos t-\sin t\end{pmatrix},\qquad a>0.}
$$
Exact secular <resonance> occurs at $a=0$: the forcing frequency matches the undamped homogeneous mode. For positive $a$, damping prevents secular growth, and the late-time periodic amplitude is proportional to $1/a$. Thus there is a resonant small-damping limit, but no unbounded resonant response at fixed $a>0$.
Since $(1-e^{-at})/a\to t$ for each fixed $t$, the requested limit gives
$$
\boxed{z(t)=t\begin{pmatrix}2\cos t\\\cos t-\sin t\end{pmatrix}\quad(a=0).}
$$
Equivalently $\alpha=c t e^{-it}$ at zero damping; its linear prefactor displays the resonance explicitly.
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