1. **State the problem:** We need to evaluate the limit $$\lim_{x\to 4} \frac{\sqrt{x+5} - 3}{x - 4}$$ and round the result to the nearest thousandth. 2. **Direct substitution:** Substitute $x=4$: $$\frac{\sqrt{4+5} - 3}{4 - 4} = \frac{\sqrt{9} - 3}{0} = \frac{3 - 3}{0} = \frac{0}{0}$$ which is an indeterminate form. 3. **Rationalize the numerator:** Multiply numerator and denominator by the conjugate of numerator: $$\frac{\sqrt{x+5} - 3}{x - 4} \cdot \frac{\sqrt{x+5} + 3}{\sqrt{x+5} + 3} = \frac{(\sqrt{x+5} - 3)(\sqrt{x+5} + 3)}{(x-4)(\sqrt{x+5} + 3)}$$ 4. **Simplify numerator:** Using difference of squares: $$ = \frac{(x+5) - 9}{(x-4)(\sqrt{x+5} + 3)} = \frac{x - 4}{(x-4)(\sqrt{x+5} + 3)}$$ 5. **Cancel $(x-4)$ terms:** $$ = \frac{1}{\sqrt{x+5} + 3}$$ 6. **Evaluate the limit:** Substitute $x=4$: $$ = \frac{1}{\sqrt{4+5} + 3} = \frac{1}{3 + 3} = \frac{1}{6} \approx 0.167$$ **Final answer:** $$\boxed{0.167}$$