Material Tensile Failure (Mechanical & Civil Engineering) Solution

Step-by-Step Solution

1. Left-sided limit ($x \to 10^-$): Approaching from the elastic/plastic deformation zone prior to fracture ($x < 10$), the stress climbs linearly:

   $$\lim_{x\to 10^-} S(x) = \lim_{x\to 10^-}(8x) = 8(10) = 80\text{ MPa}$$

2. Right-sided limit ($x \to 10^+$): Approaching from the post-fracture region ($x \ge 10$), the severed rod can bear no load:

   $$\lim_{x\to 10^+} S(x) = \lim_{x\to 10^+}(0) = 0\text{ MPa}$$

3. Limit existence and structural interpretation: Comparing the one-sided limits:

   $$\lim_{x\to 10^-} S(x) = 80\text{ MPa} \quad \neq \quad \lim_{x\to 10^+} S(x) = 0\text{ MPa}$$
   Because the one-sided limits do not match, the two-sided limit Does Not Exist (DNE).

Engineering Implication: The failure of the limit corresponds to an instantaneous drop in load capacity from $80\text{ MPa}$ to $0\text{ MPa}$ at the fracture point $x = 10\text{ mm}$. In civil and structural engineering, identifying such jump discontinuities is essential for defining safe allowable elongation limits and safety factors before ultimate tensile failure occurs.