1. **State the problem:** We are asked to find the limits of the function $f(x)$ as $x$ approaches $-2$ and $2$ from the left and right, and the overall limit at these points, based on the graph description. 2. **Recall limit definitions:** - The right-hand limit $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$. - The left-hand limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from values less than $a$. - The limit $\lim_{x \to a} f(x)$ exists if and only if the left-hand and right-hand limits are equal. 3. **Analyze limits at $x = -2$:** - From the graph, at $x = -2$, there is an open circle at $y=0$ and a filled circle at $y=2$. - The line approaching from the left (values less than $-2$) goes towards the filled circle at $y=2$. - The line approaching from the right (values greater than $-2$) goes towards the open circle at $y=0$. Therefore: $$\lim_{x \to -2^-} f(x) = 2$$ $$\lim_{x \to -2^+} f(x) = 0$$ Since these are not equal, the overall limit does not exist: $$\lim_{x \to -2} f(x) \text{ does not exist}$$ 4. **Analyze limits at $x = 2$:** - At $x=2$, there is a filled circle at $y=-2$ and an open circle at $y=2$. - The line approaching from the left (values less than $2$) goes towards the filled circle at $y=-2$. - The line approaching from the right (values greater than $2$) goes towards the open circle at $y=2$. Therefore: $$\lim_{x \to 2^-} f(x) = -2$$ $$\lim_{x \to 2^+} f(x) = 2$$ Since these are not equal, the overall limit does not exist: $$\lim_{x \to 2} f(x) \text{ does not exist}$$ **Final answers:** (a) $\lim_{x \to -2^+} f(x) = 0$ (b) $\lim_{x \to -2^-} f(x) = 2$ (c) $\lim_{x \to -2} f(x)$ does not exist (d) $\lim_{x \to 2^+} f(x) = 2$ (e) $\lim_{x \to 2^-} f(x) = -2$ (f) $\lim_{x \to 2} f(x)$ does not exist