Given the graph of a function $f$, find a number $\epsilon >0$ such that if $|x-1|<\epsilon$, then $\left|f\right(x)-1|<0.2$.

• A. $\epsilon =0.05$
• B. $\epsilon =0.3$
• C. $\epsilon =0.1$
• D. $\epsilon =0.5$

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find a value of \( \epsilon > 0 \) such that if \( |x - 1| < \epsilon \), then \( |f(x) - 1| < 0.2 \). This is related to the concept of continuity and the definition of limits.
2. Step 2: Analyze the graph of \( f(x) \). Look at the behavior of \( f(x) \) near \( x = 1 \). Specifically, observe how \( f(x) \) changes as \( x \) approaches 1 and determine the range of \( x \) values that keep \( |f(x) - 1| < 0.2 \).
3. Step 3: Determine the interval around \( x = 1 \) where \( |f(x) - 1| < 0.2 \). This involves finding the values of \( x \) such that \( f(x) \) stays within 0.2 units of 1. The interval will help you identify the corresponding \( \epsilon \).
4. Step 4: Relate the interval to \( \epsilon \). The condition \( |x - 1| < \epsilon \) defines the distance from \( x = 1 \). Choose the largest \( \epsilon \) that satisfies \( |f(x) - 1| < 0.2 \) for all \( x \) within the interval.
5. Step 5: Verify your choice of \( \epsilon \). Ensure that for the chosen \( \epsilon \), the inequality \( |f(x) - 1| < 0.2 \) holds true for all \( x \) satisfying \( |x - 1| < \epsilon \). This confirms the correctness of your solution.