Robot Arm Pivot Deflection (Entertainment & Mechanical Engineering) Solution

Step-by-Step Solution

1. Direct substitution check: Substitute $t = 2$ directly into the deflection formula:

   $$D(2) = \frac{\sqrt{2 + 7} - 3}{2 - 2} = \frac{\sqrt{9} - 3}{0} = \frac{3 - 3}{0} = \frac{0}{0}$$
   Direct substitution yields the indeterminate form $\frac{0}{0}$, meaning the deflection value cannot be computed without resolving the removable discontinuity.

2. Radical conjugate rationalization: Multiply both the numerator and denominator by the conjugate of the numerator, $(\sqrt{t + 7} + 3)$:

   $$\lim_{t\to 2} D(t) = \lim_{t\to 2} \left[ \frac{\sqrt{t + 7} - 3}{t - 2} \cdot \frac{\sqrt{t + 7} + 3}{\sqrt{t + 7} + 3} \right]$$

   Expand the numerator using the difference of squares identity $(a - b)(a + b) = a^2 - b^2$:

   $$(\sqrt{t + 7} - 3)(\sqrt{t + 7} + 3) = (\sqrt{t + 7})^2 - 3^2 = (t + 7) - 9 = t - 2$$

   Rewrite the limit with the expanded numerator and factored denominator:

   $$\lim_{t\to 2} \frac{t - 2}{(t - 2)(\sqrt{t + 7} + 3)}$$

   Cancel the common factor $(t - 2)$ for $t \neq 2$:

   $$\lim_{t\to 2} \frac{1}{\sqrt{t + 7} + 3}$$

   Evaluate by substituting $t = 2$:

   $$\frac{1}{\sqrt{2 + 7} + 3} = \frac{1}{\sqrt{9} + 3} = \frac{1}{3 + 3} = \frac{1}{6}\text{ mm} \approx 0.167\text{ mm}$$

Conclusion: The instantaneous deflection of the robotic arm at the transition point is $\frac{1}{6}\text{ mm}$ ($\approx 0.167\text{ mm}$). Rationalizing the conjugate removes the numerical singularity, providing precise structural deflection data for programming stage automation safety limits.