1. **State the problem:** We are asked to find the left-hand limit, right-hand limit, two-sided limit, and the function value at $x = -1$ for the function $f(x)$ based on the given graph. 2. **Analyze the left-hand limit $\lim_{x \to -1^-} f(x)$:** From the graph, as $x$ approaches $-1$ from the left, the function values approach 0. This matches choice A: $\lim_{x \to -1^-} f(x) = 0$. 3. **Analyze the right-hand limit $\lim_{x \to -1^+} f(x)$:** From the graph, as $x$ approaches $-1$ from the right, the function values also approach 0. 4. **Analyze the two-sided limit $\lim_{x \to -1} f(x)$:** Since both left-hand and right-hand limits equal 0, the two-sided limit exists and equals 0. 5. **Evaluate the function value $f(-1)$:** The graph shows a point at $(-1,0)$, so $f(-1) = 0$. **Summary:** - $\lim_{x \to -1^-} f(x) = 0$ - $\lim_{x \to -1^+} f(x) = 0$ - $\lim_{x \to -1} f(x) = 0$ - $f(-1) = 0$