$\text{ Let }f\text{ be a function with two continuous derivatives and f}\left(0\right)=0,{\mathrm{f}}^{\text{'}}\left(0\right)=0\text{ . Define }$a function g by $g\left(x\right)=\left\{\begin{array}{c}\frac{f\left(x\right)}{x},x\ne 0\\ 0,x=0\end{array}\right.$

Then which of the following statements is/are correct?

$g\left(x\right) is continuous at x=0 , \sin ce \underset{x\to 0}{\lim }g\left(x\right)=\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x}=\underset{x\to 0}{\lim }\frac{f^{\text{'}}\left(x\right)}{1}=f^{\text{'}}\left(0\right)=0=g\left(0\right)$

  $g\left(x\right) is is differentiable at x=0 , \sin ce g^{\text{'}}\left(0\right)=\underset{x\to 0}{\lim }\frac{g\left(x\right)-g\left(0\right)}{x-0}=\underset{x\to 0}{\lim }\frac{{\displaystyle \frac{f\left(x\right)}{x}}-0}{x}=\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^2}=\underset{x\to 0}{\lim }\frac{f\text{'}\left(x\right)}{2x}=\underset{x\to 0}{\lim }\frac{f\text{'}\text{'}\left(x\right)}{2}=\frac{f\text{'}\text{'}\left(0\right)}{2}$