1. **State the problem:** We are asked to find the limits of the function $f(x)$ as $x$ approaches 4 from the left and right, the overall limit at $x=4$, and the value of $f(4)$ based on the graph. 2. **Recall limit definitions:** - The left-hand limit $\lim_{x \to 4^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches 4 from values less than 4. - The right-hand limit $\lim_{x \to 4^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches 4 from values greater than 4. - The limit $\lim_{x \to 4} f(x)$ exists only if both left and right limits exist and are equal. - The function value $f(4)$ is the actual value of the function at $x=4$. 3. **Analyze the graph:** - From the left side, the curve approaches $y=4$ and there is a hollow circle at $(4,4)$, meaning $f(x)$ approaches 4 but $f(4)$ is not 4 from this side. - From the right side, the curve jumps to $y=6$ with a filled dot at $(4,6)$, so $f(4)=6$. 4. **Evaluate each limit and value:** - (a) $\lim_{x \to 4^-} f(x) = 4$ - (b) $\lim_{x \to 4^+} f(x) = 6$ - (c) Since left and right limits are not equal, $\lim_{x \to 4} f(x)$ does not exist. - (d) $f(4) = 6$ (value at the filled dot) **Final answers:** - (a) 4 - (b) 6 - (c) Does Not Exist - (d) 6