1. **Stating the problem:** We are asked to find the left-hand limit $\lim_{x \to 1^-} f(x)$, the right-hand limit $\lim_{x \to 1^+} f(x)$, the overall limit $\lim_{x \to 1} f(x)$, and the value of the function at $x=1$, i.e., $f(1)$, based on the graph of $f(x)$. 2. **Understanding limits:** The left-hand limit $\lim_{x \to 1^-} f(x)$ is the value that $f(x)$ approaches as $x$ approaches 1 from values less than 1. The right-hand limit $\lim_{x \to 1^+} f(x)$ is the value that $f(x)$ approaches as $x$ approaches 1 from values greater than 1. The overall limit $\lim_{x \to 1} f(x)$ exists only if both one-sided limits are equal. 3. **Analyzing the graph near $x=1$:** - From the left side ($x \to 1^-$), the graph spikes sharply upward and reaches a peak at $y=8$. - From the right side ($x \to 1^+$), the graph also spikes sharply upward and reaches the same peak at $y=8$. 4. **Determining the limits:** - $\lim_{x \to 1^-} f(x) = 8$ - $\lim_{x \to 1^+} f(x) = 8$ Since both one-sided limits are equal, the overall limit exists: $$\lim_{x \to 1} f(x) = 8$$ 5. **Determining $f(1)$:** - The graph shows a filled circle at $x=1$ at $y=1$ (the curve dips down immediately after the peak). - Therefore, $f(1) = 1$ **Final answers:** $$\lim_{x \to 1^-} f(x) = 8$$ $$\lim_{x \to 1^+} f(x) = 8$$ $$\lim_{x \to 1} f(x) = 8$$ $$f(1) = 1$$