Thermal Runaway in Electrical Discharges (Electrical Engineering) Solution

Part 1: Directional Approach & Sign Analysis

As $V \to 12^+$, the voltage stress approaches $12\text{ kV}$ from above ($V > 12$, e.g., $V = 12.001\text{ kV}$).

• Numerator: The constant value is $-2.5 < 0$ (negative).
• Denominator: Since $V > 12$, $V - 12 > 0$. The denominator approaches $0$ through positive values ($0^+$).

Part 2: Analytical Limit Calculation

Applying the signed zero rule:

Part 3: Engineering Significance

The unbounded negative surge ($-\infty$) models rapid reverse polarity ionization and thermal avalanche breakdown across the insulator. In substation engineering, this vertical asymptote defines the irreversible dielectric failure boundary; protection relays must trigger solid-state arc-flash breakers well before the system operates within the asymptotic neighborhood of $V = 12\text{ kV}$.