To solve this problem, we need to find a number $\delta$ such that:

$|x - 1| < \delta \quad \text{implies} \quad |f(x) - 1| < 0.2.$

Steps:

1. Identify the condition: The condition $|f(x) - 1| < 0.2$ means that the values of $f(x)$ must lie within the interval $(1 - 0.2, 1 + 0.2) = (0.8, 1.2)$.

2. Check the corresponding values of $x$: From the graph, we need to find the $x$-values such that $f(x)$ lies within this range. That is, we want to find the range of $x$ where $0.8 \leq f(x) \leq 1.2$.

   • From the graph, we observe that:
     • When $x = 0.7$, $f(x) \approx 1.2$.
     • When $x = 1.1$, $f(x) \approx 0.8$.

3. Calculate $\delta$: The value of $\delta$ is the distance from $x = 1$ to the boundary points of the interval $[0.7, 1.1]$.

   • So, $\delta$ is the minimum of $|1 - 0.7|$ and $|1 - 1.1|$.
   • $|1 - 0.7| = 0.3$ and $|1 - 1.1| = 0.1$.

Thus, the smallest $\delta$ that satisfies the condition is:

$\delta = 0.1.$

Would you like a more detailed explanation, or do you have any questions?

Related Questions:

1. What does the $\delta$-$\epsilon$ definition of a limit mean?
2. How can we find the value of $\delta$ for other limits?
3. What happens if $\epsilon$ is made smaller?
4. How do graphical approaches differ from algebraic approaches in limit problems?
5. Can we use this approach for discontinuous functions?

Tip:

Always carefully examine the range of values on both axes when solving limit problems graphically. This will help you determine intervals for both $x$ and $f(x)$.