For the function shown below, find $($if the quantity exists$)($A$)\underset{x{0}^{+}}{lim}f(x),($B$)\underset{x{0}^{-}}{lim}f(x),($C$)\underset{x0}{lim}f(x),$ and $($D$)f(0).$

$f(x)=\{\begin{array}{ccc}6-x^2,& \text{f}\text{o}\text{r}& x0\\ 6+x^2,& \text{f}\text{o}\text{r}& x>0\end{array}$