= Solution
Retain the exact damping factor and expand only the slowly accumulating phase. With $T=\epsilon t$ bounded,
$$
\omega t=t-\tfrac12\epsilon^2t+O(\epsilon^4t),\qquad
\frac{\sin(\omega t)}\omega=\sin t-\tfrac12\epsilon^2t\cos t+O(\epsilon^2).
$$
The resulting <two-time expansion of a weakly damped linear oscillator> is
$$
\boxed{z(\epsilon,t)=e^{-\epsilon t}\left(\sin t-\frac{\epsilon^2t}{2}\cos t\right),\qquad x-z=O(\epsilon^2)\quad(\epsilon t=O(1)).}
$$
In slow-time notation the second term in parentheses is $-\epsilon T\cos t/2$, an order-$\epsilon$ correction. Simply retaining $e^{-\epsilon t}\sin t$ would have order-$\epsilon$ rather than order-$\epsilon^2$ error on this scale, and would not meet the requested accuracy.
When $\epsilon^2t=O(1)$, the phase shift is order one: Taylor-expanding its sine and cosine is no longer an ordered asymptotic expansion of the oscillatory factor. The exact solution and $z$ acquire visibly different phases after removing their common exponential damping. In contrast, for $\epsilon t=O(1)$ the main difference between $y$ and $z$ is their amplitude envelope: $1-\epsilon t$ versus $e^{-\epsilon t}$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-79-linear-oscillator.png]
{title=Fixed-time and slow-time linear oscillator approximations, with a damping-normalized late-time phase comparison}
\b[The phrase “not valid” on the later scale needs an error convention.] For fixed positive damping, the absolute displacement of both $x$ and $z$ is exponentially small when $\epsilon^2t$ is a positive order-one number. Indeed Taylor's theorem gives, for small positive $\epsilon$,
$$
|x-z|\le C e^{-\epsilon t}\left(\epsilon^2+\epsilon^4t+\epsilon^4t^2\right).
$$
The supremum of the right side over $t\ge0$ is $O(\epsilon^2)$. Thus $z$ loses phase accuracy on $t=O(\epsilon^{-2})$, but does not lose an absolute $O(\epsilon^2)$ error bound there. The last plot displays $e^{\epsilon t}x$ and $e^{\epsilon t}z$ to show that phase failure honestly; multiplying by the damping factor would make the difference extremely small. Relative pointwise error also becomes singular near zeros, so accuracy of the oscillatory factor is the meaningful late-time comparison.
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