Answer
$$\lim_{x\to0}g(x)=4$$
Work Step by Step
$$\lim_{x\to0}\Big(\frac{4-g(x)}{x}\Big)=1$$
Here we cannot apply Quotient Rule because in the denominator, $\lim_{x\to0}x=0$.
Instead, we would start with $g(x)$: $$g(x)=-(-g(x))=-(4-g(x)-4)=-\Big(\frac{4-g(x)}{x}\times x-4\Big)$$
Therefore, $$\lim_{x\to0}g(x)=\lim_{x\to0}\Big[-\Big(\frac{4-g(x)}{x}\times x-4\Big)\Big]$$
$$\lim_{x\to0}g(x)=-\Big[\lim_{x\to0}\Big(\frac{4-g(x)}{x}\Big)\times\lim_{x\to0}(x)-\lim_{x\to0}4\Big]$$
$$\lim_{x\to0}g(x)=-(1\times0-4)=-(-4)=4$$
