Answer
The function is continuous when $a=4.$
Work Step by Step
$\lim\limits_{x\to0^-}g(x)=4[\lim\limits_{x\to0^-}\dfrac{\sin{x}}{x}]=4(1)=4.$
$\lim\limits_{x\to0^+}g(x)=a-2(0^+)=a.$
For the function to be continuous $\lim\limits_{x\to0^-}g(x)=\lim\limits_{x\to0^+}g(x)\to$
$a=4.$
Note: while $\lim\limits_{x\to0^-}\dfrac{\sin{x}}{x}$ was not studied, we know that $\lim\limits_{x\to0}\dfrac{\sin{x}}{x}=1$ and since this limit exists, this guarantees the existence of both one-sided limits and it also guarantees them being equal to $1.$
