This graph asks to find a number $\delta$ such that for $0 < |x - 3| < \delta$, it satisfies the condition $|f(x) - 2| < 0.5$. This problem deals with the concept of limits, specifically the $\epsilon-\delta$ definition of a limit.

Step-by-step approach:

1. Understanding the condition $|f(x) - 2| < 0.5$:

   • This inequality implies that the function values $f(x)$ should stay within the range $1.5 < f(x) < 2.5$ (since $2 - 0.5 = 1.5$ and $2 + 0.5 = 2.5$).

2. Identifying the $x$-range for $f(x)$ in $1.5 < f(x) < 2.5$:

   • Looking at the graph, the function reaches the value $f(x) = 2$ at $x = 3$.
   • From the graph, observe that when $f(x)$ is between 1.5 and 2.5, the corresponding $x$-values seem to range from approximately $2.6$ to $3.8$.

3. Determining $\delta$:

   • The condition $0 < |x - 3| < \delta$ means that the value of $x$ is within a distance $\delta$ from 3.
   • From the graph, we see that the function values are between 1.5 and 2.5 when $x$ is between 2.6 and 3.8.
   • Therefore, $\delta$ would be the smallest of the distances from 3 to 2.6 or 3 to 3.8.
   • The distances are $|3 - 2.6| = 0.4$ and $|3 - 3.8| = 0.8$. Hence, the smallest $\delta$ is $0.4$.

Conclusion:

The value of $\delta$ that satisfies the condition is $\delta = 0.4$.

Would you like a more detailed breakdown of the concept or have any further questions?

Here are five relative questions that can expand your understanding:

1. How does the $\epsilon-\delta$ definition of limits work in general?
2. What is the significance of the $\epsilon$ value in these problems?
3. How would the result change if we asked for $|f(x) - 2| < 0.1$?
4. What are the key differences between left-hand and right-hand limits?
5. How do we handle $\epsilon-\delta$ problems where the function is not continuous?

Tip: Always visualize or sketch the neighborhood of the limit point when dealing with $\epsilon-\delta$ problems—it can greatly help with interpreting the conditions.