Use the given graph of f to find a number 𝛿 such that
if 
0 < |x − 3| < 𝛿    then    |f(x) − 2| < 0.5.

To find a number \( \delta \) such that:

\[
0 < |x - 3| < \delta \quad \text{implies} \quad |f(x) - 2| < 0.5,
\]

we need to analyze the behavior of the function \( f(x) \) near \( x = 3 \) using the graph of \( f \). The condition means that \( f(x) \) is close to 2 when \( x \) is close to 3.

### Steps:

1. **Find the range of \( f(x) \) for \( |f(x) - 2| < 0.5 \):**
   The inequality \( |f(x) - 2| < 0.5 \) can be rewritten as:
   \[
   1.5 < f(x) < 2.5
   \]
   So, we are interested in values of \( x \) where \( f(x) \) stays between 1.5 and 2.5.

2. **Find the corresponding \( x \)-interval around \( x = 3 \):**
   Look at the graph of \( f(x) \) and find where \( f(x) \) stays within the range \( (1.5, 2.5) \). Let's say the function stays within this range for values of \( x \) between \( x = 3 - \delta \) and \( x = 3 + \delta \).

3. **Determine \( \delta \):**
   From the graph, you need to estimate the value of \( \delta \). This is the distance from \( x = 3 \) such that for all \( x \) within this interval (i.e., \( (3 - \delta, 3 + \delta) \)), the value of \( f(x) \) remains between 1.5 and 2.5.

If you provide the graph or describe its behavior near \( x = 3 \), I can help you find the exact value of \( \delta \).

Use the given graph of f to find a number 𝛿 such that if 0 < |x − 3| < 𝛿 then |f(x) − 2| < 0.5.

Learn how to find a number 𝛿 such that if 0 < |x − 3| < 𝛿, then |f(x) − 2| < 0.5 using the graph of a function. This problem explores limits, continuity, and inequalities, and applies the epsilon-delta definition of limits. Suitable for advanced high school and college students studying calculus.

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