= Noncommuting damping and observation-time limits at resonance
<Noncommuting damping and observation-time limits at resonance> means that the late-time approximation of a damped circuit is not uniform as its dissipation tends to zero. A precise comparison needs an observable: the unnormalized resonant <sinusoid> itself has no ordinary limit as time tends to infinity. Use the series <RLC circuit> with zero <initial conditions>, source $V_0\cos\omega_0t$, and $\omega_0=1/\sqrt{LC}$. For $R>0$, its steady-state <electric charge> and <electric current> are
$$
q_{\rm ss}=\frac{V_0}{R\omega_0}\sin\omega_0t,\qquad
i_{\rm ss}=\frac{V_0}{R}\cos\omega_0t.
$$
The remaining <electrical resistance> sets the resonant <amplitude>, as in the DOE series-resonance model.<DOE Fundamentals Handbook Electrical Science>[Volume 3, Module 8, printed p. 19, Series Resonance]{ref}
Define a dimensionless state error relative to that <amplitude>:
$$
D_R(t)=\sqrt{\left(\frac{Ri(t)}{V_0}-\cos\omega_0t\right)^2+
\left(\frac{R\omega_0q(t)}{V_0}-\sin\omega_0t\right)^2}.
$$
For each fixed $R>0$, the <homogeneous solution> decays and $D_R(t)\to0$. Conversely, on a fixed finite observation interval the actual circuit solution converges to the finite undamped solution as $R\downarrow0$. The two terms containing $R$ then vanish, giving $D_R(t)\to1$. Thus the genuine iterated limits are
$$
\lim_{R\downarrow0}\lim_{t\to\infty}D_R(t)=0,\qquad
\lim_{t\to\infty}\lim_{R\downarrow0}D_R(t)=1.
$$
This states the failure without treating an oscillating quantity as though it had a time limit.
For weak <damping>, put $\gamma=R/(2L)$ and $\omega_d=\sqrt{\omega_0^2-\gamma^2}$. The exact zero-state charge
$$
q(t)=\frac{V_0}{R\omega_0}\left[\sin\omega_0t-\frac{\omega_0}{\omega_d}e^{-\gamma t}\sin\omega_dt\right]
$$
both verifies the decaying transient and recovers the undamped resonant response in <undamped resonant forcing has no bounded sinusoidal steady state> by a fixed-time expansion.
For an original timing comparison, $L=10\,\mathrm{mH}$ gives envelope decay times $2L/R=0.1\,\mathrm s$ at $R=0.2\,\Omega$ and $1\,\mathrm s$ at $R=0.02\,\Omega$. At a fixed $50\,\mathrm{ms}$ the respective exponential multipliers are about $0.607$ and $0.951$. A smaller <electrical resistance> can make the sinusoidal late-time approximation worse over a fixed acquisition window. MIT's steady-state notes explicitly require the startup transient to have disappeared.<6.200 Notes: Sinusoidal Steady State by Karl K. Berggren>[Printed p. 3, late-time transient assumption]{ref}
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