= Solution
At fixed $t$, expand the exact <damped harmonic oscillator> solution using $e^{-\epsilon t}=1-\epsilon t+O(\epsilon^2)$, $\omega=1+O(\epsilon^2)$ and $1/\omega=1+O(\epsilon^2)$. Its <Poincaré expansion> through first order is
$$
\boxed{y(\epsilon,t)=(1-\epsilon t)\sin t,\qquad x(t)-y(\epsilon,t)=O(\epsilon^2)\quad(t\text{ bounded}).}
$$
The correction $-\epsilon t\sin t$ is a <secular term>. When $\epsilon t$ is order one it is not small, and the truncated amplitude eventually changes sign and grows in magnitude, instead of reproducing positive damping. This loss of ordering is why a fixed-time <regular perturbation> is unsuitable on the damping timescale.
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