1. The problem asks to find the limits of the function $f(x)$ as $x$ approaches several points where the function has discontinuities. 2. Recall that 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, if it exists. 3. Since the function has many discontinuities, we must consider the left-hand limit $\lim_{x \to a^-} f(x)$ and the right-hand limit $\lim_{x \to a^+} f(x)$ at each point $a$. If these two limits are equal, the limit exists and equals that value; otherwise, the limit does not exist. 4. From the graph description: - At $x = -2$, the function appears continuous or has a defined limit; assume $\lim_{x \to -2} f(x) = L_1$. - At $x = 0$, there is a discontinuity; check left and right limits. - At $x = 2$, similarly check left and right limits. - At $x = 4, 5, 6, 8, 10, 11$, the graph shows vertical jumps or arrows indicating discontinuities. 5. Without exact function values, we interpret the graph's behavior: - $\lim_{x \to -2} f(x) = f(-2)$ if continuous. - $\lim_{x \to 0} f(x)$ does not exist if left and right limits differ. - Similarly for other points. 6. Since the problem does not provide explicit function values or formulas, the limits at these points are determined by the graph's behavior: - $\lim_{x \to -2} f(x) = $ value at $x=-2$ (assumed continuous). - $\lim_{x \to 0} f(x)$ does not exist due to jump. - $\lim_{x \to 2} f(x)$ does not exist due to jump. - $\lim_{x \to 4} f(x)$ does not exist due to vertical jump. - $\lim_{x \to 5} f(x)$ does not exist due to vertical jump. - $\lim_{x \to 6} f(x)$ does not exist due to vertical jump. - $\lim_{x \to 8} f(x)$ does not exist due to vertical jump. - $\lim_{x \to 10} f(x)$ does not exist due to vertical jump. - $\lim_{x \to 11} f(x)$ does not exist due to vertical jump. 7. Therefore, the only limit that likely exists is at $x = -2$; all others do not exist due to discontinuities. Final answers: $$\lim_{x \to -2} f(x) = f(-2)$$ $$\lim_{x \to 0} f(x) \text{ does not exist}$$ $$\lim_{x \to 2} f(x) \text{ does not exist}$$ $$\lim_{x \to 4} f(x) \text{ does not exist}$$ $$\lim_{x \to 5} f(x) \text{ does not exist}$$ $$\lim_{x \to 6} f(x) \text{ does not exist}$$ $$\lim_{x \to 8} f(x) \text{ does not exist}$$ $$\lim_{x \to 10} f(x) \text{ does not exist}$$ $$\lim_{x \to 11} f(x) \text{ does not exist}$$