1. **Problem statement:** We need to find a number $a$ such that $\lim_{x \to a} g(x)$ exists but $g(a)$ is not defined. 2. **Understanding the problem:** The limit $\lim_{x \to a} g(x)$ exists if the left-hand limit $\lim_{x \to a^-} g(x)$ and the right-hand limit $\lim_{x \to a^+} g(x)$ both exist and are equal. 3. **Key rule:** For the limit to exist at $a$, the function values approaching $a$ from both sides must approach the same number, regardless of whether $g(a)$ is defined. 4. **Analyzing the graph:** - At $x=2$, the function $g(2)$ is not defined (open circle), but the left and right limits both exist and are equal. - At $x=4$, $g(4)$ is defined (solid point), so it does not satisfy the condition of $g(a)$ being undefined. 5. **Conclusion:** The correct answer for part (b) is $a=2$ because the limit exists but $g(2)$ is not defined. 6. **Why the initial answer was marked incorrect:** Possibly a misunderstanding or misinterpretation of the graph or the problem statement. The correct reasoning shows $a=2$ satisfies the condition. **Final answer:** $a=2$