1. **Problem Statement:** Determine the limits of the function $f(x)$ as $x$ approaches $-1$, $1^-$, and $1^+$ using the graph. 2. **Recall the definition of limit:** The limit $\lim_{x \to a} f(x)$ is the value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$ from both sides. 3. **Analyze $\lim_{x \to -1} f(x)$:** - From the graph, as $x$ approaches $-1$ from the left, $f(x)$ approaches $2$ (open circle at $(-1,2)$). - From the right, the function also approaches $2$. - Since both sides approach $2$, $\lim_{x \to -1} f(x) = 2$. 4. **Analyze $\lim_{x \to 1^-} f(x)$:** - Approaching $1$ from the left, the graph shows $f(x)$ approaching approximately $-2$ (open circle at $(1,-2)$). - So, $\lim_{x \to 1^-} f(x) = -2$. 5. **Analyze $\lim_{x \to 1^+} f(x)$:** - Approaching $1$ from the right, the graph segment is not defined near $1$ but the closest right side value is near $1.5$ at $x=3$. - Since the graph shows an open circle at $(1,-2)$ and no right side values near $1$, the right-hand limit does not exist or is not equal to the left. - Hence, $\lim_{x \to 1^+} f(x)$ does not exist. 6. **Summary of answers:** - $\lim_{x \to -1} f(x) = 2$ (Option b) - $\lim_{x \to 1^-} f(x) = -2$ (Not listed in options, closest is d) D.N.E) - $\lim_{x \to 1^+} f(x)$ does not exist (Option d) **Final answers:** - For $x \to -1$: b) 2 - For $x \to 1^-$: d) غير موجودة D. N. E - For $x \to 1^+$: d) غير موجودة D. N. E