1. **State the problem:** We need to find the limit $$\lim_{x \to \infty} \frac{\ln x}{2x}$$. 2. **Recall the behavior of functions:** As $x \to \infty$, $\ln x$ grows without bound but very slowly compared to $x$, which grows much faster. 3. **Apply limit properties:** The function is a ratio of $\ln x$ (numerator) and $2x$ (denominator). Since $x$ grows faster than $\ln x$, the fraction should approach zero. 4. **Use L'Hôpital's Rule to confirm:** Since both numerator and denominator approach infinity, we differentiate numerator and denominator: $$\lim_{x \to \infty} \frac{\ln x}{2x} = \lim_{x \to \infty} \frac{\frac{d}{dx}(\ln x)}{\frac{d}{dx}(2x)} = \lim_{x \to \infty} \frac{\frac{1}{x}}{2} = \lim_{x \to \infty} \frac{1}{2x}$$ 5. **Evaluate the simplified limit:** As $x \to \infty$, $\frac{1}{2x} \to 0$. 6. **Final answer:** $$\boxed{0}$$ Thus, $$\lim_{x \to \infty} \frac{\ln x}{2x} = 0$$.