= Decay of a damped oscillator under exponentially decaying forcing
For $x''+2\gamma x'+(\gamma^2+\mu^2)x=g(t)$ with $\gamma,\mu>0$, zero initial data give $x(t)=\mu^{-1}\int_0^te^{-\gamma(t-s)}\sin(\mu(t-s))g(s)\,ds$. If $|g(s)|\leq Ce^{-as}$ for $a>0$, the <causal Green function of a damped oscillator> gives
$$
|x(t)|\leq\frac C\mu\begin{cases}\dfrac{e^{-at}-e^{-\gamma t}}{\gamma-a},&a\neq\gamma,\\te^{-\gamma t},&a=\gamma.\end{cases}
$$
Both bounds tend to zero. A decaying homogeneous transient can be added for arbitrary finite initial data. The coincident decay rates require the extra polynomial factor and cannot be handled by dividing by zero in the first expression.
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