1. **Problem Statement:** Given the piecewise linear function $f(x)$ with specified points and limits, find the values of $f(1)$ and the limits of $f(x)$ as $x$ approaches 1, 2, and 3 from the left and right. 2. **Recall the definitions:** - The value of the function at a point $x=a$ is $f(a)$. - The left-hand limit $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from values less than $a$. - The right-hand limit $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$. - The limit $\lim_{x \to a} f(x)$ exists if and only if the left-hand and right-hand limits are equal. 3. **Evaluate $f(1)$:** - From the graph, the solid point at $x=1$ is at $y=3$. - Therefore, $f(1) = 3$. 4. **Evaluate $\lim_{x \to 1^-} f(x)$:** - Approaching 1 from the left, the graph approaches the open circle at $(1,2)$. - So, $\lim_{x \to 1^-} f(x) = 2$. 5. **Evaluate $\lim_{x \to 1^+} f(x)$:** - Approaching 1 from the right, the graph starts at the solid point $(1,3)$. - So, $\lim_{x \to 1^+} f(x) = 3$. 6. **Evaluate $\lim_{x \to 1} f(x)$:** - Since $\lim_{x \to 1^-} f(x) = 2$ and $\lim_{x \to 1^+} f(x) = 3$ are not equal, the limit does not exist. 7. **Evaluate $\lim_{x \to 2^-} f(x)$:** - Approaching 2 from the left, the graph approaches the open circle at $(2,3)$. - So, $\lim_{x \to 2^-} f(x) = 3$. 8. **Evaluate $\lim_{x \to 2^+} f(x)$:** - Approaching 2 from the right, the graph approaches the open circle at $(2,2)$. - So, $\lim_{x \to 2^+} f(x) = 2$. 9. **Evaluate $\lim_{x \to 2} f(x)$:** - Since $\lim_{x \to 2^-} f(x) = 3$ and $\lim_{x \to 2^+} f(x) = 2$ are not equal, the limit does not exist. 10. **Evaluate $\lim_{x \to 3^-} f(x)$:** - Approaching 3 from the left, the graph approaches the solid point at $(3,1)$. - So, $\lim_{x \to 3^-} f(x) = 1$. 11. **Evaluate $\lim_{x \to 3^+} f(x)$:** - Approaching 3 from the right, the graph approaches the open circle at $(3,1)$. - So, $\lim_{x \to 3^+} f(x) = 1$. 12. **Evaluate $\lim_{x \to 3} f(x)$:** - Since $\lim_{x \to 3^-} f(x) = 1$ and $\lim_{x \to 3^+} f(x) = 1$ are equal, the limit exists and equals 1. **Final answers:** - $f(1) = 3$ - $\lim_{x \to 1^-} f(x) = 2$ - $\lim_{x \to 1^+} f(x) = 3$ - $\lim_{x \to 1} f(x)$ does not exist - $\lim_{x \to 2^-} f(x) = 3$ - $\lim_{x \to 2^+} f(x) = 2$ - $\lim_{x \to 2} f(x)$ does not exist - $\lim_{x \to 3^-} f(x) = 1$ - $\lim_{x \to 3^+} f(x) = 1$ - $\lim_{x \to 3} f(x) = 1$