Audio Pre-Amplifier Hard Limiting (Electrical & Audio Engineering)

In live sound and broadcast engineering, a high-gain audio pre-amplifier circuit boosts an incoming microphone input voltage $v$ (in Volts). To prevent power amplifier overload and speaker cone burnout during acoustic feedback spikes, the amplified signal passes through a multi-stage non-linear compression and hard-limiting saturation circuit.

The total output voltage $V_{\text{out}}(v)$ as a function of the input surge voltage $v$ is modeled by the composite function:

As a sudden electrostatic discharge or feedback loop causes the microphone input voltage to surge toward infinity ($v \to \infty$):

1. Use the multi-stage inside-out composite limit technique to evaluate $\lim_{v \to \infty} V_{\text{out}}(v)$.
2. Determine whether the output signal clamps at a finite voltage threshold, and state the maximum output voltage delivered to downstream audio speakers.