1. **Problem Statement:** Calculate the following limits for the piecewise function $f(x)$ based on the given graph: a) $\lim_{x \to 3^+} f(x)$ b) $\lim_{x \to 3^-} f(x)$ c) $\lim_{x \to 2^+} f(x)$ d) $\lim_{x \to 2^-} f(x)$ e) $\lim_{x \to -1^+} f(x)$ f) $\lim_{x \to -1^-} f(x)$ g) $\lim_{x \to -3} f(x)$ 2. **Key Concept:** 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. - The right-hand limit $\lim_{x \to a^+} f(x)$ considers values of $x$ approaching $a$ from the right (greater than $a$). - The left-hand limit $\lim_{x \to a^-} f(x)$ considers values of $x$ approaching $a$ from the left (less than $a$). 3. **Evaluate each limit using the graph description:** a) $\lim_{x \to 3^+} f(x)$: For $x$ just greater than 3, $f(x)$ decreases linearly from $(3,2)$ to $(4,1)$, so the limit is the value at $x=3$ from the right side, which approaches 2. b) $\lim_{x \to 3^-} f(x)$: For $x$ just less than 3, $f(x)$ increases linearly from $(2,1)$ to $(3,3)$, so the limit is the value approaching 3 from the left, which is 3. c) $\lim_{x \to 2^+} f(x)$: For $x$ just greater than 2, $f(x)$ increases linearly from $(2,1)$ to $(3,3)$, so the limit is 1. d) $\lim_{x \to 2^-} f(x)$: For $x$ just less than 2, $f(x)$ decreases linearly from $(1,1)$ to $(2,1)$, so the limit is 1. e) $\lim_{x \to -1^+} f(x)$: For $x$ just greater than -1, $f(x)$ decreases linearly from $(0,1)$ to $(1,1)$, so the limit at $-1^+$ is the value at $x=-1$ from the right side, which is 1. f) $\lim_{x \to -1^-} f(x)$: For $x$ just less than -1, $f(x)$ is constant at 3 from $-2$ to $-1$, so the limit is 3. g) $\lim_{x \to -3} f(x)$: From the left, $f(x)$ is constant at 1 approaching $-3$; from the right, $f(x)$ is constant at 2 approaching $-3$. Since the left and right limits differ, the limit does not exist. 4. **Summary of limits:** $$\lim_{x \to 3^+} f(x) = 2$$ $$\lim_{x \to 3^-} f(x) = 3$$ $$\lim_{x \to 2^+} f(x) = 1$$ $$\lim_{x \to 2^-} f(x) = 1$$ $$\lim_{x \to -1^+} f(x) = 1$$ $$\lim_{x \to -1^-} f(x) = 3$$ $$\lim_{x \to -3} f(x) \text{ does not exist}$$