Smart Grid Dynamic Power Clamping (Computer & Electrical Engineering)

In smart power distribution grid substations, an embedded microcontroller manages dynamic power routing $P(I)$ (in Kilowatts) across a transformer busbar as a function of incoming line current $I$ (in Amperes).

To prevent arithmetic division-by-zero faults during sensor polling while ensuring power flow remains seamless across a calibration point at $I = 4\text{ A}$ , the power management firmware defines the piecewise transfer function:

P(I) = \begin{cases} \frac{I^2 - 16}{I - 4} & : I \neq 4 \\ k I - 2 & : I = 4 \end{cases}

Evaluate the limit of the unpatched power expression as current approaches the threshold: $\lim_{I \to 4} \frac{I^2 - 16}{I - 4}$ .
Using the three-part continuity rubric, determine the exact parameter value $k$ required to make $P(I)$ continuous for all current levels $I$ .
Explain how setting this exact $k$ value prevents electrical relay tripping in the smart grid substation.