For function  $f\left(x\right)=\frac{\ln x}{x}$  which of the following statement(s) is/are correct?

Given $f\left(x\right)=\frac{\ln x}{x}$

Differentiate both sides with respect to x

$\begin{array}{l}f\text{'}\left(x\right)=\frac{x.\frac{1}{x}-\ln x}{x^2}=\frac{1-\ln x}{x^2}\\ \text{for horizontal tangent }\ln x=1\Rightarrow x=e\\ If\text{\hspace{0.17em}cuts the }x-axisthen\frac{\ln x}{x}=0\Rightarrow x=1\\ f^{\text{'}}\left(x\right)\text{\hspace{0.17em} is positive in }x\in \left(0,e\right)andf\text{'}\left(x\right)is-ve if x is more than e\\ f\left(x\right)\text{\hspace{0.17em} is monotonic, }\\ It is many to one function \\ \text{for vertical tangent }f\text{'}\left(x\right)=\infty \\ \Rightarrow \frac{1-\ln x}{x^2}=\infty \Rightarrow \frac{x^2}{1-\ln x}=0\\ x=0\text{ which is not in domain}\end{array}$