Boundary Failure Classification (Aerospace & Civil Engineering)

A structural systems engineer is analyzing three different stress, pressure, and electrical simulation scenarios near critical system boundaries, modeled by the functions below:

1. Shockwave Boundary (Aerospace/Mechanical): The localized gas pressure $P(x)$ (in atmospheres) near a supersonic nozzle shock boundary at $x = 0$ is modeled by:
   $$P(x) = \frac{|x|}{x}$$
   Evaluate the limit $\lim_{x \to 0} P(x)$.
2. Stress Singularity (Civil/Structural): The mechanical stress $\sigma(x)$ (in MPa) in a steel girder near a micro-crack singularity at $x = 2\text{ mm}$ is modeled by:
   $$\sigma(x) = \frac{5}{(x - 2)^2}$$
   Evaluate the limit $\lim_{x \to 2} \sigma(x)$.
3. Signal Chatter (Electrical/Instrumentation): A sensor's signal voltage response $V(t)$ (in mV) under high-frequency electrical interference near $t = 0\text{ seconds}$ is modeled by:
   $$V(t) = \sin\left(\frac{1}{t}\right)$$
   Evaluate the limit $\lim_{t \to 0} V(t)$.

For each scenario, determine if the limit exists. If the limit does not exist (DNE), classify the failure using one of the three structural behaviors (Jump Discontinuity, Unbounded Behavior, or Wildly Oscillating Behavior) and state the engineering implications.