Rocket Payload Velocity (Mechanical & Aerospace Engineering)

An aerospace engineer is analyzing the launch trajectory of a multi-stage sounding rocket during atmospheric ascent. The instantaneous velocity $V(t)$ (in $\text{m/s}$) depends on the engine thrust factor $T(t)$ (in $\text{m/s}^2$) and the normalized propellant mass burn factor $M(t)$ relative to burn-time $t$ (in seconds), governed by $V(t) = T(t) \cdot M(t)$.

Telemetry data confirms the following limits as the rocket approaches the end of first-stage burn at $t \to 4\text{ seconds}$:

• $\lim_{t\to 4} T(t) = 8\text{ m/s}^2$
• $\lim_{t\to 4} M(t) = 12$

1. Identify the Limit Law needed to evaluate $\lim_{t\to 4} V(t)$.
2. Calculate the payload velocity limit $\lim_{t\to 4} V(t)$.