From the given graph, we need to find a value of $\delta$ such that if $|x - 1| < \delta$, then $|f(x) - 1| < 0.2$.

Key conditions to understand:

1. $|f(x) - 1| < 0.2$ means that the function value $f(x)$ must lie between 0.8 and 1.2 (since 1 - 0.2 = 0.8 and 1 + 0.2 = 1.2).
2. $|x - 1| < \delta$ defines the range around $x = 1$ such that the function values satisfy the condition for $f(x)$.

Steps:

1. Observe the graph around $x = 1$. We can see the behavior of $f(x)$ as it moves away from $x = 1$.
2. When $x = 1$, $f(x) = 1$, so we are looking for how far $x$ can deviate from 1 such that $f(x)$ stays within the bounds 0.8 and 1.2.

From the graph:

• At $x \approx 0.9$, $f(x)$ is close to 1.2.
• At $x \approx 1.1$, $f(x)$ is close to 0.8.

Thus, for $f(x)$ to stay between 0.8 and 1.2, $\delta$ is approximately 0.1, since $|x - 1| < 0.1$ keeps $f(x)$ within the desired range.

Conclusion:

$\delta = 0.1$.

Let me know if you have further questions or need more details! Here are some related questions to explore:

1. How would the value of $\delta$ change if the condition were $|f(x) - 1| < 0.1$?
2. What is the formal definition of the epsilon-delta criterion in limits?
3. Can you calculate the slope of the curve $f(x)$ at $x = 1$?
4. How would you find the value of $\delta$ if the graph was more complex?
5. How do we use limits to formally prove continuity?

Tip: When dealing with limit problems, always check the graph for critical points where the function crosses specific bounds, as this helps in determining appropriate $\delta$ values.