Final answer:
The question is about finding the delta (δ) corresponding to specified epsilon (ε) values for the limit of a function as x approaches 2, which involves working backwards from the definition of a limit using algebraic manipulations.
Step-by-step explanation:
Step-by-step explanation:
To find the corresponding values of Δ (delta) for given values of ε (epsilon), we need to determine the largest possible range of x-values within which the function will stay within ε (epsilon) distance of the limit.
For ε = 0.2:
Substitute ε = 0.2 into the equation |x - 2| < ε to get |x - 2| < 0.2.
Solve the inequality to find the range of x: -0.2 < x - 2 < 0.2.
Add 2 to each part of the inequality to get 1.8 < x < 2.2.
Therefore, the largest possible value of Δ for ε = 0.2 is 0.2.
For ε = 0.1:
Substitute ε = 0.1 into the equation |x - 2| < ε to get |x - 2| < 0.1.
Solve the inequality to find the range of x: -0.1 < x - 2 < 0.1.
Add 2 to each part of the inequality to get 1.9 < x < 2.1.
Therefore, the largest possible value of Δ for ε = 0.1 is 0.1.
The mathematical concept being asked about involves finding the δ (δ-delta) that corresponds to a given ε (ε-epsilon) for the limit of a function as x approaches a specific value. Specifically, the limit is x → 2 for the function f(x) = x^3 - 4x + 3. When ε = 0.2 and ε = 0.1, we are tasked with finding a δ value such that for all x within δ of 2, the value of f(x) is within ε of the limit, which is 3. This type of problem is typically solved by working backwards from the definition of limit and possibly making use of inequalities or solving equations.
Considering the function given, we would find an appropriate δ by ensuring that |f(x) - L| < ε where L is the limit value. We would substitute the values of x that result in the difference f(x) - L being less than ε using algebraic manipulations to find the largest possible δ that satisfies the condition for the corresponding ε.