Robot Arm Pivot Deflection (Entertainment & Mechanical Engineering)

A mechatronics engineer designing an animatronic stage prop for an entertainment production is evaluating the transient dynamic tip deflection $D(t)$ (in millimeters) of a robotic arm during a sudden high-speed rotation. The deflection is modeled as a function of elapsed time $t$ (in seconds) by:

The motion controller requires the exact theoretical deflection limit at the pivot transition timestamp $t = 2\text{ seconds}$.

1. Verify that direct substitution at $t = 2$ produces an indeterminate form.
2. Use radical conjugate rationalization to compute $\lim_{t\to 2} D(t)$.