Evaluate a limit at infinity using L'Hopital's Rule

As $x \to \infty$, $\ln(x) \to \infty$ and $x \to \infty$, so the limit has the indeterminate form $\infty/\infty$. This justifies applying L'Hopital's Rule, which requires differentiating the numerator and denominator separately: $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$ and $\frac{d}{dx}[x] = 1$.

Applying the rule gives $\lim_{x \to \infty} \frac{\ln(x)}{x} = \lim_{x \to \infty} \frac{1/x}{1} = \lim_{x \to \infty} \frac{1}{x}.$ As $x \to \infty$, $1/x \to 0$, so the original limit equals $0$.