Terminal Velocity of a Re-entry Capsule (Aerospace Engineering)

An aerospace engineering team modeling the atmospheric re-entry descent of an unmanned scientific capsule determines that its downward velocity $v(t)$ (in meters per second) as a function of descent time $t$ (in seconds) is governed by the rational exponential model:

To verify that the drogue parachutes and main deceleration canopies can safely deploy without exceeding maximum dynamic pressure and tensile tear limits, the recovery team must determine the capsule's asymptotic terminal velocity ($v_{\text{term}} = \lim_{t \to \infty} v(t)$).

1. Identify the dominant growth terms as $t \to \infty$.
2. Evaluate the limit of the capsule's velocity analytically as $t \to \infty$:
   $$\lim_{t \to \infty} \frac{76e^{0.4t} - 12}{2e^{0.4t} + 5}$$
3. State the horizontal asymptote equation and interpret its value for parachute structural design.