Complete study of (ln x)/x

Limits. As $x\to0^+$: $\ln x\to-\infty$, $\frac1x\to+\infty$, so $f\to-\infty$: the line $x=0$ is a vertical asymptote. As $x\to+\infty$: $f\to0$ (growth comparison), so $y=0$ is a horizontal asymptote. The only zero is $x=1$.

Variations. $f'>0$ for $x<e$ and $f'<0$ for $x>e$: increasing on $(0,e]$, decreasing on $[e,\infty)$. Maximum $f(e)=\frac1e$.

Convexity. $f''(x)=\dfrac{2\ln x-3}{x^3}$: concave on $(0,e^{3/2}]$, convex on $[e^{3/2},\infty)$, inflection point at $x=e^{3/2}\approx4.48$, where $f=\frac3{2e^{3/2}}\approx0.335$.

The inequality. $f(x)\le f(e)=\frac1e$, i.e. $\ln x\le\frac xe$ for $x>0$ (equality at $x=e$). It is the tangent line of $\ln$ at $e$.

Remark. $f(a)=f(b)$ with $a<b$ forces $a<e<b$. For positive integers, the only pair $a<b$ with $a^b=b^a$ is $(2,4)$, as $f(2)=f(4)=\frac{\ln2}2$ (indeed $2^4=4^2=16$). (For negative integers, $a^b=b^a$ is not equivalent to $f(a)=f(b)$, since $\ln$ is not defined there.)