$\text{ Let }f\left(x\right)\text{ be defined for all }x>0\text{ and be continuous. Let }$$f\left(x\right)\text{ satisfies }f\left(\frac{x}{y}\right)=f\left(x\right)-f\left(y\right)\text{ for all }x,y\text{ and }f\left(e\right)= 1. Then$

• A

  $f\left(x\right)\text{ is bounded }$

• B

  $f\left(\frac{1}{x}\right)\to 0\text{ as }x\to 0$

• C

  $xf\left(x\right)\to 1\text{ as }x\to 0$

• D

  $f\left(x\right)={\log }_{e}x$