1. **State the problem:** We need to find the limit $$\lim_{x \to -2} \frac{g(x)}{h(x)}$$ where functions $g$ and $h$ are given graphically. 2. **Analyze $g(x)$ near $x = -2$:** From the description, $g(x)$ has an open circle at $(-2,5)$ and a closed circle at $(-2,1)$. The limit depends on the value $g(x)$ approaches as $x$ approaches $-2$, not the value at $x=-2$ itself. 3. **Determine $\lim_{x \to -2} g(x)$:** Since the graph approaches the open circle at $(-2,5)$ from the left and right, the limit is $5$. 4. **Analyze $h(x)$ near $x = -2$:** The graph passes through $(-2,1)$ continuously. 5. **Determine $\lim_{x \to -2} h(x)$:** Since $h$ is continuous at $x=-2$, the limit is $1$. 6. **Calculate the limit:** $$ \lim_{x \to -2} \frac{g(x)}{h(x)} = \frac{\lim_{x \to -2} g(x)}{\lim_{x \to -2} h(x)} = \frac{5}{1} = 5 $$ 7. **Answer:** The limit is $5$, which corresponds to choice D.