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Let's go through the problem you've provided step by step.

### Part (a)
The problem gives a formal definition of the limit, and we're focusing on the function:

\[
\left| \sqrt{13 - x} - 2 \right| < \epsilon \quad \text{whenever} \quad |x - 9| < \delta
\]

This is a typical epsilon-delta proof scenario. The limit of \( f(x) = \sqrt{13 - x} \) as \( x \to 9 \) is being evaluated. Based on the function, we expect:

\[
\lim_{x \to 9} \sqrt{13 - x} = 2
\]

So, the formal definition is verifying that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - 9| < \delta \), we have \( \left| \sqrt{13 - x} - 2 \right| < \epsilon \).

### Part (b)
We are tasked with finding \( \delta \) when \( \epsilon = 1 \), and showing the steps.

1. Start with the inequality \( \left| \sqrt{13 - x} - 2 \right| < \epsilon \). Set \( \epsilon = 1 \):
   \[
   \left| \sqrt{13 - x} - 2 \right| < 1
   \]

2. We need to solve this inequality to express \( \delta \) in terms of \( x \). First, remove the absolute value by breaking the inequality into two cases:
   \[
   -1 < \sqrt{13 - x} - 2 < 1
   \]

3. Add 2 to all parts of the inequality:
   \[
   1 < \sqrt{13 - x} < 3
   \]

4. Square all parts of the inequality to get rid of the square root:
   \[
   1^2 < 13 - x < 3^2
   \]
   Simplifying this, we get:
   \[
   1 < 13 - x < 9
   \]

5. Subtract 13 from all parts of the inequality:
   \[
   -12 < -x < -4
   \]

6. Multiply everything by \( -1 \), which flips the inequality signs:
   \[
   4 < x < 12
   \]

7. Therefore, \( x \) must be within the range \( 4 < x < 12 \). This means the distance from \( x = 9 \) must be less than 5, since \( 9 - 4 = 5 \) and \( 12 - 9 = 3 \). Hence, we conclude that:

   \[
   \delta = 3
   \]

Thus, \( \delta = 3 \) when \( \epsilon = 1 \).

Would you like more details on any part of this explanation?

### Related Questions:
1. How do epsilon-delta proofs work for other types of functions?
2. Can \( \delta \) depend on \( \epsilon \) in more complex functions?
3. How can we handle cases where the function involves more complicated expressions, like rational functions?
4. What happens if we change the value of \( \epsilon \) to something smaller, like \( \epsilon = 0.5 \)?
5. How can this technique be extended to multivariable limits?

### Tip:
When solving inequalities involving square roots, always check if squaring introduces extraneous solutions, especially for negative values.

The formal limit definition, “for every ε > 0, there exists a δ > 0 such that |√13−x−2| < ε whenever |x−9| < δ”, defines the limit equation lim(x → 9) √13−x = 2. Find δ when ε = 1. Show the steps of computation.

Learn how to solve an epsilon-delta problem for the limit of √13 − x as x approaches 9. This detailed solution shows step-by-step how to find δ when ε = 1 using inequalities and the epsilon-delta limit definition. Ideal for college-level calculus students.

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