Given the table of values for \(f(x)\): \(f(1) = 2\), \(f(1.5) = 2.5\), \(f(1.9) = 2.9\), \(f(0.5) = 1.5\), estimate \(\lim_{x \to 1} f(x)\) using values approaching from both sides.

To estimate \(\lim_{x \to 1} f(x)\) using the given values, we will look at the function values as \(x\) approaches 1 from both the left and the right. From the table, we have the following values: - \(f(1) = 2\) - \(f(1.5) = 2.5\) (approaching from the right) - \(f(1.9) = 2.9\) (further approaching from the right) - \(f(0.5) = 1.5\) (approaching from the left) - \(f(1.5)\) and \(f(1.9)\) indicate that as \(x\) increases past 1, \(f(x)\) is increasing. Now, let's analyze the values: 1. **Approaching from the left**: - The closest value we have is \(f(0.5) = 1.5\). However, we don't have a value closer to 1 from the left. We can infer that as \(x\) approaches 1 from the left, \(f(x)\) is likely increasing towards \(f(1)\). 2. **Approaching from the right**: - The values \(f(1.5) = 2.5\) and \(f(1.9) = 2.9\) show that as \(x\) approaches 1 from the right, \(f(x)\) is also increasing and is greater than \(f(1)\). Given that \(f(1) = 2\) is the value at \(x = 1\), and the values approaching from both sides suggest that \(f(x)\) is increasing as we move away from 1, we can conclude: - As \(x\) approaches 1 from the left, \(f(x)\) is likely approaching 2. - As \(x\) approaches 1 from the right, \(f(x)\) is also approaching 2. Thus, we can estimate: \[ \lim_{x \to 1} f(x) = 2 \] This limit is consistent from both sides, confirming that the limit exists and is equal to 2.