Use the graph of  f(x)=x2
  below to find a number  δ
  such that  ∣∣x2−1∣∣<0.2
  whenever  0<|x−1|<δ
 .

We are tasked with finding a number \( \delta \) such that the following inequality holds:

\[
|x^2 - 1| < 0.2 \quad \text{whenever} \quad 0 < |x - 1| < \delta.
\]

This is essentially a problem of finding the appropriate \( \delta \) to satisfy the given conditions based on the behavior of the function \( f(x) = x^2 \).

### Step-by-Step Solution:

1. **Rewrite the condition**: 
   We need to find \( \delta \) such that if \( |x - 1| < \delta \), then \( |x^2 - 1| < 0.2 \). This can be interpreted as an \(\varepsilon-\delta\) definition of continuity at \( x = 1 \), with \( \varepsilon = 0.2 \).

2. **Simplify \( |x^2 - 1| \)**:
   Notice that:
   \[
   |x^2 - 1| = |(x - 1)(x + 1)|.
   \]
   So the inequality becomes:
   \[
   |(x - 1)(x + 1)| < 0.2.
   \]

3. **Bound \( |x + 1| \)**:
   Since we are interested in \( x \) values close to \( 1 \), we can make an estimate. If \( |x - 1| < \delta \), then \( x \) is close to \( 1 \). Specifically, we know that \( x \) will be in the range \( (1 - \delta, 1 + \delta) \), so:
   \[
   |x + 1| \approx 2 \quad \text{when} \quad x \text{ is near } 1.
   \]
   This is a rough approximation for \( x \) values close to \( 1 \).

4. **Solve for \( \delta \)**:
   Using the approximation \( |x + 1| \approx 2 \), the inequality \( |(x - 1)(x + 1)| < 0.2 \) becomes:
   \[
   |x - 1| \cdot 2 < 0.2,
   \]
   which simplifies to:
   \[
   |x - 1| < \frac{0.2}{2} = 0.1.
   \]

Thus, we find that \( \delta = 0.1 \) should work. That is, if \( |x - 1| < 0.1 \), then \( |x^2 - 1| < 0.2 \).

### Conclusion:
The number \( \delta \) that satisfies the condition is \( \delta = 0.1 \).

Would you like more details, or do you have any questions? Here are some follow-up questions:

1. How can the \(\varepsilon-\delta\) definition of continuity be applied to this problem?
2. What would happen if \( \varepsilon = 0.1 \) instead of \( 0.2 \)?
3. Why is it necessary to approximate \( |x + 1| \) when solving for \( \delta \)?
4. Can the value of \( \delta \) be larger than \( 0.1 \) for some other choice of approximation?
5. How does the graph of \( f(x) = x^2 \) help visualize the solution?

**Tip:** When solving \(\varepsilon-\delta\) problems, a good first step is often to rewrite expressions in factored form to make the relationship between variables clearer.

Use the graph of f(x) = x^2 below to find a number δ such that |x^2 − 1| < 0.2 whenever 0 < |x − 1| < δ.

Learn how to find the value of δ such that |x^2 − 1| < 0.2 when 0 < |x − 1| < δ, based on the ε-δ definition of continuity. This problem covers key calculus concepts, including limits and quadratic functions. Suitable for advanced high school and college students.

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