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:

\lim_{V \to 12^{+}} \frac{-2.5}{V - 12} = \frac{-2.5}{0^+} = -\infty

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 $V = 12\text{ kV}$ .