Structural Resonance Threshold (Civil Engineering)

A structural engineer modeling dynamic wind shear on a suspended pedestrian walkway determines that the structural vibration amplitude $A(\omega)$ (in millimeters) as a function of the wind vortex shedding frequency $\omega$ (in radians per second) is given by:

The walkway structure possesses an un-damped natural resonance frequency at $\omega_0 = 8\text{ rad/s}$. To size dynamic tuned mass dampers, the engineering team must determine how the vibration amplitude escalates as the wind excitation frequency increases from lower values and approaches this critical resonance threshold.

1. Classify the behavior of $A(\omega)$ as $\omega \to 8^-$ using directional sign analysis.
2. Evaluate the limit analytically:
   $$\lim_{\omega \to 8^{-}} \frac{15}{64 - \omega^2}$$
3. State the physical significance of this mathematical behavior for structural integrity.