1. **State the problem:** We are given a piecewise graph of a function $f$ and need to find the values of various limits and function values at $x=2$ and $x=4$. 2. **Recall limit definitions:** - 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 limit $\lim_{x \to a} f(x)$ exists if and only if the left and right limits are equal. - The function value $f(a)$ is the value of the function at $x=a$. 3. **Analyze the graph at $x=2$:** - From the left, the curve approaches the open circle at $(2,1)$, so $\lim_{x \to 2^-} f(x) = 1$. - From the right, the curve starts at the filled point $(2,3)$, so $\lim_{x \to 2^+} f(x) = 3$. - Since left and right limits differ, $\lim_{x \to 2} f(x)$ does not exist. - The function value at $x=2$ is the filled point $f(2) = 3$. 4. **Analyze the graph at $x=4$:** - The curve is continuous and smooth near $x=4$, so $\lim_{x \to 4} f(x)$ is the $y$-value of the curve at $x=4$. - From the description, the curve rises to about $(4.3,4)$, so at $x=4$ the value is approximately $3.9$ (since it is rising slightly before $4.3$). - The function value $f(4)$ is on the curve, so $f(4) \approx 3.9$. **Final answers:** (a) $\lim_{x \to 2^-} f(x) = 1$ (b) $\lim_{x \to 2^+} f(x) = 3$ (c) $\lim_{x \to 2} f(x)$ does not exist (d) $f(2) = 3$ (e) $\lim_{x \to 4} f(x) \approx 3.9$ (f) $f(4) \approx 3.9$