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 $(3,4)$. This means: - The limit of $g(x)$ as $x$ approaches 3 is the value at the open circle, which is $3$. - The function value $g(3)$ is $4$ (closed circle), but this does not affect the limit. 3. **Analyze the behavior of $f(x)$ near $x=3$:** The graph of $f$ is continuous with a point at $(3,0)$. - So, $\lim_{x \to 3} f(x) = f(3) = 0$. 4. **Evaluate the limit:** - Numerator limit: $\lim_{x \to 3} g(x) = 3$ - Denominator limit: $\lim_{x \to 3} f(x) = 0$ Since the denominator approaches zero and numerator approaches a nonzero number, the fraction $\frac{g(x)}{f(x)}$ tends to infinity or negative infinity depending on the sign of $f(x)$ near 3. 5. **Check the sign of $f(x)$ near 3:** - From the graph description, $f(x)$ moves downward to $(3,0)$ from $(2,4)$ and then upward to $(4,4)$. - This suggests $f(x)$ approaches zero from positive values on both sides. 6. **Conclusion:** - The limit $$\lim_{x \to 3} \frac{g(x)}{f(x)}$$ does not exist because the denominator approaches zero while numerator approaches a nonzero number, causing the fraction to grow without bound. **Final answer:** The limit does not exist.