1. **State the problem:** We need to find the left-hand limit, right-hand limit, two-sided limit, and the function value at $x = -2$ for the function $f(x)$ based on the graph. 2. **Recall limit definitions:** - The left-hand limit $\lim_{x \to -2^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $-2$ from the left. - The right-hand limit $\lim_{x \to -2^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $-2$ from the right. - The two-sided limit $\lim_{x \to -2} f(x)$ exists only if both one-sided limits are equal. - The function value $f(-2)$ is the value of the function at $x = -2$. 3. **Analyze the graph near $x = -2$:** - From the left side ($x \to -2^-$), the graph approaches the point $(-2, -2)$ with a solid dot, so the left-hand limit is $-2$. - From the right side ($x \to -2^+$), the graph continues downward from about $y = -2$ to $y = -6$ near $x = -1$, so the right-hand limit is approximately $-6$. 4. **Evaluate the limits:** $$\lim_{x \to -2^-} f(x) = -2$$ $$\lim_{x \to -2^+} f(x) = -6$$ 5. **Check if two-sided limit exists:** Since $-2 \neq -6$, the two-sided limit does not exist. 6. **Find $f(-2)$:** The graph shows a solid dot at $(-2, -2)$, so $$f(-2) = -2$$ **Final answers:** $$\lim_{x \to -2^-} f(x) = -2$$ $$\lim_{x \to -2^+} f(x) = -6$$ $$\lim_{x \to -2} f(x) = \text{DNE}$$ $$f(-2) = -2$$