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Electrical Surge Protection Switch (Computer & Electrical Engineering)

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A computer hardware engineer designs an automated over-voltage surge protection circuit for an embedded server power supply. The electric current profile I(t)I(t) (in amperes) relative to initialization time tt (in milliseconds) exhibits a rapid switching threshold at t=2 mst = 2\text{ ms}:

I(t)={4:t<2t2+5:t≥2I(t) = \begin{cases} 4 & : t < 2 \\ t^2 + 5 & : t \ge 2 \end{cases}

  1. Evaluate the left-sided current limit lim⁡t→2−I(t)\lim_{t\to 2^-} I(t).
  2. Evaluate the right-sided current limit lim⁡t→2+I(t)\lim_{t\to 2^+} I(t).
  3. Determine whether the overall limit lim⁡t→2I(t)\lim_{t\to 2} I(t) exists, and explain what the result indicates about the electrical power transition.