1. **Problem Statement:** We analyze the function $g$ and find values of $a$ that satisfy different limit and function value conditions based on the graph. 2. **Recall Limit and Function Value Definitions:** - The limit $\lim_{x \to a} g(x)$ exists if the left-hand limit $\lim_{x \to a^-} g(x)$ and right-hand limit $\lim_{x \to a^+} g(x)$ both exist and are equal. - The function value $g(a)$ is defined if there is a point on the graph at $x=a$. 3. **Part (a):** $\lim_{x \to a} g(x)$ does not exist but $g(a)$ is defined. - At $a=4$, the graph shows a defined point $g(4)$ (closed circle) but the limits from left and right differ (discontinuity). - So, $a=4$ satisfies this. 4. **Part (b):** $\lim_{x \to a} g(x)$ exists but $g(a)$ is not defined. - At $a=2$, the limit does not exist because left and right limits differ (open circle and jump), so this is incorrect. 5. **Part (c):** $\lim_{x \to a^-} g(x)$ and $\lim_{x \to a^+} g(x)$ both exist but $\lim_{x \to a} g(x)$ does not exist. - This means left and right limits exist but are not equal. - At $a=2$ and $a=4$, both one-sided limits exist but differ, so both satisfy this. 6. **Part (d):** $\lim_{x \to a^+} g(x) = g(a)$ but $\lim_{x \to a^-} g(x) \neq g(a)$. - At $a=4$, the right-hand limit equals the function value but the left-hand limit differs. - So, $a=4$ satisfies this. **Final answers:** - (a) $a=4$ - (b) No (given $a=2$ is incorrect) - (c) $a=2$ and $a=4$ - (d) $a=4$