Delta Finding Ed25C3

1. **State the problem:** We need to find $\delta$ such that whenever $0 < |x - 3| < \delta$, it follows that $|f(x) - 2| < 0.5$. 2. **Understand the inequality:** The condition $|f(x) - 2| < 0.5$ means that $f(x)$ lies between $1.5$ and $2.5$. 3. **Analyze the graph:** From the graph, the values of $x$ for which $f(x)$ is between $1.5$ and $2.5$ are approximately between $2.6$ and $3.8$. 4. **Calculate $\delta$:** Since $x$ is centered at $3$, the distance to the endpoints of this interval are: $$3 - 2.6 = 0.4$$ $$3.8 - 3 = 0.8$$ 5. **Choose the minimum distance:** To ensure $|f(x) - 2| < 0.5$ for all $x$ within $\delta$ of $3$, we take the smaller of these two distances: $$\delta = \min(0.4, 0.8) = 0.4$$ 6. **Final answer:** $$\boxed{\delta = 0.4}$$ This means if $0 < |x - 3| < 0.4$, then $|f(x) - 2| < 0.5$ holds true.