1. **Problem statement:** Find the limit of the function $$\frac{\ln x}{10\sqrt{x}}$$ as $$x \to \infty$$ using the derivative ratio $$\frac{g'(x)}{h'(x)}$$ where $$g(x) = \ln x$$ and $$h(x) = 10\sqrt{x}$$. 2. **Recall the derivatives:** - $$g'(x) = \frac{1}{x}$$ - $$h(x) = 10x^{\frac{1}{2}}$$ so $$h'(x) = 10 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = 5x^{-\frac{1}{2}}$$ 3. **Set up the ratio of derivatives:** $$\frac{g'(x)}{h'(x)} = \frac{\frac{1}{x}}{5x^{-\frac{1}{2}}} = \frac{1}{x} \cdot \frac{1}{5x^{-\frac{1}{2}}} = \frac{1}{5} \cdot \frac{1}{x} \cdot x^{\frac{1}{2}} = \frac{1}{5} x^{-\frac{1}{2}}$$ 4. **Simplify the expression:** $$\frac{1}{5} x^{-\frac{1}{2}} = \frac{1}{5\sqrt{x}}$$ 5. **Evaluate the limit:** As $$x \to \infty$$, $$\frac{1}{5\sqrt{x}} \to 0$$. 6. **Apply L'Hôpital's Rule:** Since $$\lim_{x \to \infty} \frac{g'(x)}{h'(x)} = 0$$, the original limit is also $$\lim_{x \to \infty} \frac{\ln x}{10\sqrt{x}} = 0$$. **Final answer:** $$\boxed{0}$$