1. **State the problem:** We are asked to find the values of $f(1)$ and the left-hand and right-hand limits of $f(x)$ as $x$ approaches 1, 2, and 3, as well as the two-sided limits at these points. 2. **Recall definitions:** - The value $f(a)$ is the function value at $x=a$. - The left-hand limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left. - The right-hand limit $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right. - The two-sided limit $\lim_{x \to a} f(x)$ exists if and only if the left-hand and right-hand limits are equal. 3. **Analyze the graph at $x=1$:** - There is a closed dot at $(1,3)$, so $f(1) = 3$. - The left-hand limit $\lim_{x \to 1^-} f(x)$ is the $y$-value approaching from the left side. The line descends to the point $(1,2)$ which is an open circle, so the left limit is $2$. - The right-hand limit $\lim_{x \to 1^+} f(x)$ is the value from the right side, which is the closed dot at $(1,3)$, so the right limit is $3$. - Since left and right limits differ, $\lim_{x \to 1} f(x)$ does not exist. 4. **Analyze the graph at $x=2$:** - There is an open circle at $(2,2)$ and a closed dot at $(2,3)$ is not mentioned, so $f(2)$ is not explicitly given but from the description, the closed dot at $(2,3)$ is at $x=2$ (from the line from open circle (1,2) to closed dot (2,3)), so $f(2) = 3$. - The left-hand limit $\lim_{x \to 2^-} f(x)$ approaches the closed dot at $(2,3)$, so left limit is $3$. - The right-hand limit $\lim_{x \to 2^+} f(x)$ approaches the open circle at $(2,2)$, so right limit is $2$. - Since left and right limits differ, $\lim_{x \to 2} f(x)$ does not exist. 5. **Analyze the graph at $x=3$:** - There is an open circle at $(3,1)$, so $f(3)$ is not defined at 1. - The left-hand limit $\lim_{x \to 3^-} f(x)$ approaches the open circle at $(3,1)$, so left limit is $1$. - The right-hand limit $\lim_{x \to 3^+} f(x)$ is from the line descending from $(3,1)$ to $(5,-1)$, starting at open circle $(3,1)$, so right limit is also $1$. - Since left and right limits are equal, $\lim_{x \to 3} f(x) = 1$. **Final answers:** $$f(1) = 3$$ $$\lim_{x \to 1^-} f(x) = 2$$ $$\lim_{x \to 1^+} f(x) = 3$$ $$\lim_{x \to 1} f(x) \text{ does not exist}$$ $$f(2) = 3$$ $$\lim_{x \to 2^-} f(x) = 3$$ $$\lim_{x \to 2^+} f(x) = 2$$ $$\lim_{x \to 2} f(x) \text{ does not exist}$$ $$f(3) \text{ is undefined}$$ $$\lim_{x \to 3^-} f(x) = 1$$ $$\lim_{x \to 3^+} f(x) = 1$$ $$\lim_{x \to 3} f(x) = 1$$