The question is asking to find a value of $\delta$ such that for all $x$, if $0 < |x - 3| < \delta$, then $|f(x) - 2| < 0.5$. This is related to the definition of continuity, specifically in terms of the epsilon-delta definition of limits.

Analyzing the graph:

• The horizontal line $y = 2$ is relevant because we are interested in how the function $f(x)$ deviates from 2.
• The condition $|f(x) - 2| < 0.5$ means we are looking for where the function lies between 1.5 and 2.5 (as $2 - 0.5 = 1.5$ and $2 + 0.5 = 2.5$).
• From the graph, this means we are interested in the interval where the graph of $f(x)$ lies between these two values of $y$.

Now, we need to determine for which values of $x$, $f(x)$ is between 1.5 and 2.5:

• At $f(2.6)$, the graph is approximately at 1.5.
• At $f(3.8)$, the graph reaches 2.5.

Therefore, $|f(x) - 2| < 0.5$ holds when $x$ is between approximately 2.6 and 3.8. Since the problem asks for $|x - 3| < \delta$, we can infer the distance from $x = 3$ to these boundary points.

Solution:

• The distance from 3 to 2.6 is $3 - 2.6 = 0.4$.
• The distance from 3 to 3.8 is $3.8 - 3 = 0.8$.

Therefore, $\delta$ can be the smaller of these two values, which is $\delta = 0.4$.

Thus, $\delta = 0.4$.

Would you like a deeper explanation of how this ties into the epsilon-delta definition of limits?

Related questions:

1. What is the epsilon-delta definition of limits?
2. How do you determine delta given epsilon in a limit problem?
3. Why is $\delta = 0.4$ chosen instead of 0.8 in this case?
4. How do continuity and limits relate to each other in calculus?
5. How can graphing help to solve limit problems visually?

Tip: In epsilon-delta limit problems, always focus on the range for $f(x)$ (based on epsilon) and find the corresponding interval for $x$ around the given point.