1. **Problem Statement:** Find a number $a$ for the function $g$ such that: (a) $\lim_{x \to a^-} g(x)$ does not exist but $g(a)$ is defined. (b) $\lim_{x \to a^-} g(x)$ exists but $g(a)$ is not defined. (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. (d) $\lim_{x \to a^+} g(x) = g(a)$ but $\lim_{x \to a^-} g(x) \neq g(a)$. 2. **Key Concepts:** - The left-hand limit $\lim_{x \to a^-} g(x)$ is the value $g(x)$ approaches as $x$ approaches $a$ from the left. - The right-hand limit $\lim_{x \to a^+} g(x)$ is the value $g(x)$ approaches as $x$ approaches $a$ from the right. - The limit $\lim_{x \to a} g(x)$ exists only if both left and right limits exist and are equal. - $g(a)$ is the actual function value at $x=a$. 3. **Step-by-step Analysis:** **(a)** $\lim_{x \to a^-} g(x)$ does not exist but $g(a)$ is defined. - From the graph, at $a=4$, the left-hand limit does not exist (due to jump or oscillation), but $g(4)$ is defined. - So, $a=4$ satisfies (a). **(b)** $\lim_{x \to a^-} g(x)$ exists but $g(a)$ is not defined. - At $a=2$, the left-hand limit exists, but $g(2)$ is defined (filled dot), so this is incorrect. **(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. - From the graph, at $a=2$ and $a=4$, both one-sided limits exist but differ, so $\lim_{x \to a} g(x)$ does not exist. - So, $a=2$ and $a=4$ satisfy (c). **(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 $g(4)$ but the left-hand limit does not equal $g(4)$. - So, $a=4$ satisfies (d). 4. **Final answers:** - (a) $a=4$ - (b) No correct $a$ given (2 is incorrect) - (c) $a=2$ and $a=4$ - (d) $a=4$