1. **Problem Statement:** Determine for which values of $x_0$ in the interval $-3 \leq x_0 \leq 2$ the limit $\lim_{x \to x_0} g(x)$ exists for the given piecewise linear function $g$. 2. **Recall the limit existence rule:** The limit $\lim_{x \to x_0} g(x)$ exists if and only if the left-hand limit $\lim_{x \to x_0^-} g(x)$ and the right-hand limit $\lim_{x \to x_0^+} g(x)$ both exist and are equal. 3. **Analyze the graph description:** - From $x=-3$ to $x=-2$, the function is continuous (solid points, no breaks). - At $x=-2$, the function has a solid point and an open circle at $x=-1$, indicating a possible jump or discontinuity. - From $x=-1$ to $x=0$, the function descends to the origin with solid points at $x=0$. - From $x=0$ to $x=1$, the function descends with a solid point at $x=1$ and an open circle just before it, indicating a jump. - From $x=1$ to $x=2$, the function rises continuously. 4. **Check points for limit existence:** - At $x=-3$ to $x=-2$: limit exists (continuous segments). - At $x=-2$: limit exists if left and right limits match (solid point and line segment suggest yes). - At $x=-1$: open circle indicates a jump, so left and right limits differ; limit does not exist. - At $x=0$: solid point and continuous segment, limit exists. - At $x=1$: open circle before solid point indicates jump; limit does not exist. - At $x=2$: endpoint with continuous segment, limit exists. 5. **Conclusion:** The limit $\lim_{x \to x_0} g(x)$ exists for all $x_0$ in $[-3,2]$ except at points where there are jumps, specifically at $x_0 = -1$ and $x_0 = 1$. **Final answer:** $$\lim_{x \to x_0} g(x) \text{ exists for all } -3 \leq x_0 \leq 2 \text{ except at } x_0 = -1 \text{ and } x_0 = 1.$$