Answer
$$\lim_{x\to0}g(x)=-\frac{1}{2}$$
Work Step by Step
$$\lim_{x\to-4}\Big(x\lim_{x\to0}g(x)\Big)=2$$
Apply Product Rule: $$\lim_{x\to-4}x\times\lim_{x\to-4}\Big(\lim_{x\to0}g(x)\Big)=2$$
$$-4\lim_{x\to-4}\Big(\lim_{x\to0}g(x)\Big)=2$$
$$\lim_{x\to-4}\Big(\lim_{x\to0}g(x)\Big)=-\frac{1}{2}$$
We can suppose that $\lim_{x\to0}g(x)$ exists and $\lim_{x\to0}g(x)=k$ ($k\in R$)
$$\lim_{x\to-4}k=-\frac{1}{2}$$
Because $k$ is a number, $\lim_{x\to-4}k=k$ $$k=-\frac{1}{2}$$
$$\lim_{x\to0}g(x)=-\frac{1}{2}$$
