Given that

$\underset{x2}{lim}f(x)=4,\underset{x2}{lim}g(x)=-4,\underset{x2}{lim}h(x)=0$

find each limit$,$ if it exists. $($If an answer does not exist, enter DNE.$)$

$($a$)\underset{x2}{lim}[f(x)+2g(x)]$

$($b$)\underset{x2}{lim}[g(x){]}^3$

$($c$)\underset{x2}{lim}\sqrt[\phantom{2}]{f(x)}$

$($d$)\underset{x2}{lim}\frac{2f(x)}{g(x)}$

$($e$)\underset{x2}{lim}\frac{g(x)}{h(x)}$

$($f$)\underset{x2}{lim}\frac{g(x)h(x)}{f(x)}$