Final answer:
The student asked about limits in mathematics, and the response involves finding the largest delta values for given epsilon tolerances for the function lim x → 1 (4x − 3) = 1. The largest deltas found for epsilon values of 0.5, 0.1, and 0.05 are 0.125, 0.025, and 0.0125 respectively.
Step-by-step explanation:
The student is asking about the concept of limits in mathematics, specifically the limit of the function as x approaches 1. We are given that lim x → 1 (4x − 3) = 1, and we need to find the largest delta (Δx) values that correspond to epsilon (ε) values of 0.5, 0.1, and 0.05, where epsilon represents the allowable error from the limit value, and delta represents the distance from the point x = 1 within which the function's value stays within the epsilon range.
- For ε = 0.5, we solve |4x - 3 - 1| < 0.5 → |4x - 4| < 0.5 → |x - 1| < 0.125. The largest delta is 0.125.
- For ε = 0.1, we solve |4x - 3 - 1| < 0.1 → |4x - 4| < 0.1 → |x - 1| < 0.025. The largest delta is 0.025.
- For ε = 0.05, we solve |4x - 3 - 1| < 0.05 → |4x - 4| < 0.05 → |x - 1| < 0.0125. The largest delta is 0.0125.