Electrical Surge Protection Switch (Computer & Electrical Engineering) Solution

Step-by-Step Solution

1. Left-sided limit ($t \to 2^-$): For times prior to the 2 ms switching threshold ($t < 2$), the circuit operates in steady baseline mode governed by $I(t) = 4$:

   $$\lim_{t\to 2^-} I(t) = \lim_{t\to 2^-}(4) = 4\text{ A}$$

2. Right-sided limit ($t \to 2^+$): For times at or after the switching event ($t \ge 2$), the circuit enters the active surge suppression state governed by $I(t) = t^2 + 5$:

   $$\lim_{t\to 2^+} I(t) = \lim_{t\to 2^+}(t^2 + 5) = 2^2 + 5 = 4 + 5 = 9\text{ A}$$

3. Two-sided limit evaluation and physical interpretation: Comparing the directional limits:

   $$\lim_{t\to 2^-} I(t) = 4\text{ A} \quad \neq \quad \lim_{t\to 2^+} I(t) = 9\text{ A}$$
   Since the left-hand and right-hand limits are not equal, the overall two-sided limit Does Not Exist (DNE).

Engineering Implication: This limit failure is a jump discontinuity with a sudden step increase of $5\text{ amperes}$ at $t = 2\text{ ms}$. This indicates a sharp switching transient, alerting the computer engineer to incorporate decoupling capacitors to suppress switching noise and protect sensitive logic components.