Limit at infinity and horizontal asymptote for f(x) = ln(x)/x

As $x \to +\infty$, both $\ln(x)$ and $x$ grow without bound, so this is an $\infty/\infty$ situation. However, $\ln(x)$ grows much more slowly than $x$: for large $x$, the denominator outpaces the numerator by an ever-increasing factor. This is analogous to comparing a slowly growing power (like $x^{1/10}$) against $x^1$ in a rational-function comparison — the higher power in the denominator dominates.

Since $x$ grows faster than $\ln(x)$ for all large $x$, the ratio $\ln(x)/x$ shrinks toward $0$. Formally, $\lim_{x\to\infty}\frac{\ln x}{x}=0$. Because this finite limit exists as $x\to+\infty$, the graph of $f$ approaches the horizontal line $y=0$, so $y=0$ is a horizontal asymptote.