1. **State the problem:** We need to find the limit $$\lim_{x \to \infty} \frac{2 \ln x}{5 x}$$. 2. **Recall the behavior of functions:** As $$x \to \infty$$, $$\ln x$$ grows without bound but much slower than any linear function like $$x$$. 3. **Apply the limit:** Since $$x$$ grows faster than $$\ln x$$, the fraction $$\frac{\ln x}{x}$$ approaches 0. 4. **Multiply by constants:** The constants 2 and 5 do not affect the limit's tendency, so $$\lim_{x \to \infty} \frac{2 \ln x}{5 x} = \frac{2}{5} \lim_{x \to \infty} \frac{\ln x}{x} = \frac{2}{5} \times 0 = 0.$$