Earthquake Damping Envelope (Civil & Structural Engineering)

During high-rise building design in seismically active regions, structural engineers install tuned mass dampers (TMDs) at the roof level to absorb vibrational energy during earthquakes.

Upon the sudden arrival of an impulsive shear wave at time $t = 0$ seconds, the initial transient lateral displacement $x(t)$ (in centimeters) of the damper assembly for $t > 0$ is governed by the high-frequency oscillatory response:

1. Set up a bounding sandwich inequality using the fundamental range of the sine function.
2. Apply the Squeeze Theorem to evaluate $\lim_{t \to 0^+} x(t)$.
3. Explain the civil engineering significance of this limit for the structural safety of the building's joints at the instant of shock impact.