Let's solve the problem step by step: ### Step 1: Find \(\lim_{x \to 2^-} f(x)\) #### **Explanation:** - The notation \(\lim_{x \to 2^-} f(x)\) means we want to find the **left-hand limit** of the function as \(x\) approaches 2. - Look at the graph and observe the behavior of the function as \(x\) approaches 2 from values less than 2 (from the left). #### **From the Graph:** - As \(x\) approaches 2 from the left, the function value oscillates up and down but gets closer and closer to \(y = 0\). - The graph has an open circle at \(x = 2\), but we are only interested in the approach, not the actual value at \(x = 2\). ### **Final Answer:** \[ \lim_{x \to 2^-} f(x) = 0 \] **Explanation:** As \(x\) approaches 2 from the left, the y-values of the function get closer to 0, so the left-hand limit is 0.

Let's revisit the problem carefully and analyze the graph step by step: ### Step 1: Find \(\lim_{x \to 2^-} f(x)\) #### **Explanation:** - The notation \(\lim_{x \to 2^-} f(x)\) refers to the **left-hand limit** of \(f(x)\) as \(x\) approaches 2 from values less than 2. - To determine this, we observe the behavior of the function on the graph as \(x\) approaches 2 from the left side. #### **From the Graph:** - As \(x\) approaches 2 from the left, the graph shows the function values approaching **-1**. - There is an open circle at \(x=2\) with \(f(x)\) approaching \(-1\) from the left, indicating the limit is \(-1\). **Important note:** Even if the function value at \(x=2\) is different or not defined, the limit depends on the behavior as \(x \to 2^-\), not the value at \(x=2\). ### **Final Answer:** \[ \boxed{\lim_{x \to 2^-} f(x) = -1} \] **Explanation:** From the graph, as \(x\) approaches 2 from the left, \(f(x)\) approaches \(-1\). Therefore, the left-hand limit is \(-1\).