1. **Problem Statement:** Evaluate the following for the function $f(x)$ based on the given graph: (a) $f(3)$ (b) $\lim_{x \to 3} f(x)$ (c) $\lim_{x \to 0^-} f(x)$ (d) $\lim_{x \to 0^+} f(x)$ (e) $\lim_{x \to 0} f(x)$ (f) $f(0)$ (g) Is $f$ continuous at $x=0$? Explain. (h) $\lim_{x \to 2} f(x)$ 2. **Key Concepts:** - The value of the function at a point $x=a$ is $f(a)$. - The limit $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from both sides. - One-sided limits $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ are limits from the left and right respectively. - A function is continuous at $x=a$ if $f(a)$ exists, $\lim_{x \to a} f(x)$ exists, and they are equal. 3. **Step-by-step Evaluation:** **(a) Evaluate $f(3)$:** From the graph, at $x=3$, the function value is approximately $2$. **(b) Evaluate $\lim_{x \to 3} f(x)$:** The graph near $x=3$ approaches $2$ from both sides, so $\lim_{x \to 3} f(x) = 2$. **(c) Evaluate $\lim_{x \to 0^-} f(x)$:** Approaching $0$ from the left, the graph is near $0$, so $\lim_{x \to 0^-} f(x) = 0$. **(d) Evaluate $\lim_{x \to 0^+} f(x)$:** Approaching $0$ from the right, the graph is also near $0$, so $\lim_{x \to 0^+} f(x) = 0$. **(e) Evaluate $\lim_{x \to 0} f(x)$:** Since left and right limits at $0$ are equal, $\lim_{x \to 0} f(x) = 0$. **(f) Evaluate $f(0)$:** From the graph, $f(0)$ is near $0$. **(g) Is $f$ continuous at $x=0$?** Since $f(0)$ exists and equals the limit $\lim_{x \to 0} f(x) = 0$, $f$ is continuous at $x=0$. **(h) Evaluate $\lim_{x \to 2} f(x)$:** At $x=2$, the graph is smooth and near $1.5$, so $\lim_{x \to 2} f(x) = 1.5$. 4. **Final Answers:** (a) $f(3) = 2$ (b) $\lim_{x \to 3} f(x) = 2$ (c) $\lim_{x \to 0^-} f(x) = 0$ (d) $\lim_{x \to 0^+} f(x) = 0$ (e) $\lim_{x \to 0} f(x) = 0$ (f) $f(0) = 0$ (g) $f$ is continuous at $x=0$ (h) $\lim_{x \to 2} f(x) = 1.5$