Final answer:
To determine the value of δ for ε = 0.1, we start with the inequality |x² - 3x + 2| < 0.1 and factor it. By constraining x around 2, we analyze the function's behavior and estimate that a δ of 0.1 is appropriate, which is an initial estimation and might be adjusted with further analysis.
Step-by-step explanation:
To find the value of δ corresponding to ε = 0.1 using the ε - δ definition of a limit for the function as x approaches 2, we want to ensure that |x² - 3x + 1 + 1| < 0.1 whenever 0 < |x - 2| < δ. This simplifies to |x² - 3x + 2| < 0.1.
Factoring the quadratic expression, we get |(x - 1)(x - 2)| < 0.1. To find δ, we need to analyze the inequality for values of x near 2.
Assuming δ is small, we can approximate it around 2, constraining x between (2 - δ) and (2 + δ). We start by examining the function at points close to 2 where the product (x - 1)(x - 2) is positive, and then find the maximum δ that ensures the product is less than 0.1.
For example, if x is within 0.1 of 2, say x = 2.1 or 1.9, then |x - 1| will be approximately 1, and |x - 2| will be 0.1, which gives us a product of 0.1. This suggests that a δ value of 0.1 would suffice in this case. This is a preliminary estimation, and exact δ computation might require more rigorous analysis or specific algebraic manipulation depending on the function's behavior near x = 2.