1. **State the problem:** Find the limit $$\lim_{b \to 6} \frac{\frac{1}{b} - \frac{1}{6}}{b - 6}$$. 2. **Recall the formula:** This is a difference quotient resembling the definition of a derivative: $$\lim_{x \to a} \frac{f(x) - f(a)}{x - a} = f'(a)$$ where here, $f(b) = \frac{1}{b}$ and $a = 6$. 3. **Simplify the numerator:** $$\frac{1}{b} - \frac{1}{6} = \frac{6 - b}{6b}$$ 4. **Rewrite the limit:** $$\lim_{b \to 6} \frac{\frac{6 - b}{6b}}{b - 6} = \lim_{b \to 6} \frac{6 - b}{6b(b - 6)}$$ 5. **Notice that $6 - b = -(b - 6)$, so:** $$\lim_{b \to 6} \frac{-(b - 6)}{6b(b - 6)}$$ 6. **Cancel the common factor $(b - 6)$:** $$\lim_{b \to 6} \frac{\cancel{-(b - 6)}}{6b\cancel{(b - 6)}} = \lim_{b \to 6} \frac{-1}{6b}$$ 7. **Evaluate the limit by substituting $b = 6$:** $$\frac{-1}{6 \times 6} = \frac{-1}{36}$$ **Final answer:** $$\boxed{\frac{-1}{36}}$$