Op-Amp Transient Response and Voltage Saturation (Electrical Engineering)

In analog circuit design, operational amplifier (op-amp) integrators and active filters exhibit bounded output dynamics constrained by internal DC supply voltage rails ($V_{CC}$). When excited by an abrupt step voltage, the output voltage response initially charges through feedback capacitive impedance, but cannot exceed the physical rail saturation voltage regardless of how much time elapses.

An electrical engineer is testing a precision audio pre-amplifier circuit. When energized at time $t = 0$ (measured in milliseconds, $t \ge 0$), the transient output response $V(t)$ (in Volts) across the feedback loop is modeled by the rational saturation function:

1. Determine the domain and identify any real discontinuities of the mathematical model for all real numbers $t$. Verify whether any vertical asymptotes occur within the physical operating window $t \ge 0$.
2. Calculate the initial output voltage at the instant of power energization $t = 0$.
3. Evaluate the horizontal asymptote of the response by computing $\lim_{t \to \infty} V(t)$ using the Horizontal Asymptote Rational Degree Rules. State the physical circuit limit represented by this horizontal asymptote.
4. Construct an algebraic sign chart for the first derivative $V'(t)$ on the domain $t \ge 0$ to verify that the voltage profile is strictly monotonic increasing toward saturation.
5. In practical instrumentation, an amplifier is considered "settled" when its output reaches $90\%$ of its steady-state saturation voltage ($0.90 \times V_{\text{sat}}$). Determine the required settling time $t_{\text{settle}}$.