$f:R^{†}\to R\text{ be a continuous function satisfying }$$f\left(\frac{x}{y}\right)=f\left(x\right)-f\left(y\right)\mathrm{\forall }x,y\in R^{+}.\text{If }{f}^{\mathrm{\prime }}\left(1\right)=1,\text{ then }$

• A

  $f \text{is unbounded }$

• B

  $\underset{x\to 0}{lim} f\left(\frac{1}{x}\right)=0$

• C

  $\underset{x\to 0}{lim} \frac{f(1+x)}{x}=1$

• D

  $\underset{x\to 0}{lim} x\cdot f\left(x\right)=0$