1. **State the problem:** Evaluate the limit $$\lim_{x \to \infty} \frac{\ln(x)}{\sqrt[3]{x}}$$ without rounding, giving an exact answer. 2. **Recall the behavior of functions:** As $x \to \infty$, $\ln(x)$ grows without bound but very slowly, while $\sqrt[3]{x} = x^{1/3}$ grows faster because it is a power function. 3. **Rewrite the limit:** $$\lim_{x \to \infty} \frac{\ln(x)}{x^{1/3}}$$ 4. **Apply L'Hôpital's Rule:** Since both numerator and denominator tend to infinity, differentiate numerator and denominator: $$\lim_{x \to \infty} \frac{\frac{d}{dx} \ln(x)}{\frac{d}{dx} x^{1/3}} = \lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{1}{3} x^{-2/3}}$$ 5. **Simplify the fraction:** $$= \lim_{x \to \infty} \frac{1/x}{\frac{1}{3} x^{-2/3}} = \lim_{x \to \infty} \frac{1}{x} \cdot \frac{3}{x^{-2/3}} = \lim_{x \to \infty} 3 \cdot \frac{1}{x} \cdot x^{2/3}$$ 6. **Combine powers of $x$:** $$= \lim_{x \to \infty} 3 x^{\cancel{-1} + \cancel{\frac{2}{3}}} = \lim_{x \to \infty} 3 x^{-\frac{1}{3}}$$ 7. **Rewrite the limit:** $$= 3 \lim_{x \to \infty} \frac{1}{x^{1/3}}$$ 8. **Evaluate the limit:** As $x \to \infty$, $x^{1/3} \to \infty$, so $\frac{1}{x^{1/3}} \to 0$. 9. **Final answer:** $$\lim_{x \to \infty} \frac{\ln(x)}{\sqrt[3]{x}} = 0$$