1. **State the problem:** We need to find the limit $$\lim_{x \to 3} \frac{g(x)}{f(x)}$$ where functions $g$ and $f$ are given graphically. 2. **Analyze the behavior of $g(x)$ near $x=3$:** From the description, $g(x)$ has an open circle at $(3,3)$ and a closed circle at $(3,4)$. The value of $g(3)$ is the closed circle, so $g(3) = 4$. However, the limit depends on the values approaching 3, which is the value at the open circle, so $$\lim_{x \to 3} g(x) = 3.$$ 3. **Analyze the behavior of $f(x)$ near $x=3$:** The graph of $f$ passes through $(3,0)$, so $$\lim_{x \to 3} f(x) = 0.$$ 4. **Evaluate the limit:** We have $$\lim_{x \to 3} \frac{g(x)}{f(x)} = \frac{\lim_{x \to 3} g(x)}{\lim_{x \to 3} f(x)} = \frac{3}{0}.$$ Division by zero indicates the limit may not exist or may be infinite. 5. **Check one-sided limits of $f(x)$ near 3:** - From the left, $f(x)$ is decreasing to 0 at $x=3$. - From the right, $f(x)$ increases from 0 to 4. Since $f(x)$ approaches 0 from different sides and $g(x)$ approaches 3, the ratio $$\frac{g(x)}{f(x)}$$ will approach $+\infty$ or $-\infty$ or not exist depending on the sign of $f(x)$ near 3. 6. **Conclusion:** Because the denominator approaches zero and numerator approaches a nonzero number, the limit does not exist. **Final answer:** The limit does not exist.