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The limit problem provided is:

\[
\lim_{{x \to b}} \frac{b - x}{\sqrt{x} - \sqrt{b}}
\]

This is an indeterminate form of the type \( \frac{0}{0} \) when \( x \to b \). To resolve this, we can multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the square roots.

### Step-by-Step Solution:

1. The original limit is:
   \[
   \lim_{{x \to b}} \frac{b - x}{\sqrt{x} - \sqrt{b}}
   \]

2. Multiply the numerator and denominator by the conjugate of the denominator:
   \[
   \frac{b - x}{\sqrt{x} - \sqrt{b}} \cdot \frac{\sqrt{x} + \sqrt{b}}{\sqrt{x} + \sqrt{b}} = \frac{(b - x)(\sqrt{x} + \sqrt{b})}{(\sqrt{x} - \sqrt{b})(\sqrt{x} + \sqrt{b})}
   \]

3. Use the difference of squares in the denominator:
   \[
   (\sqrt{x} - \sqrt{b})(\sqrt{x} + \sqrt{b}) = x - b
   \]

4. Now the expression becomes:
   \[
   \frac{(b - x)(\sqrt{x} + \sqrt{b})}{x - b}
   \]

5. Since \( b - x = -(x - b) \), the expression simplifies to:
   \[
   \lim_{{x \to b}} \frac{-(x - b)(\sqrt{x} + \sqrt{b})}{x - b} = \lim_{{x \to b}} -(\sqrt{x} + \sqrt{b})
   \]

6. As \( x \to b \), \( \sqrt{x} \to \sqrt{b} \), so the limit becomes:
   \[
   -(\sqrt{b} + \sqrt{b}) = -2\sqrt{b}
   \]

### Final Answer:
\[
\lim_{{x \to b}} \frac{b - x}{\sqrt{x} - \sqrt{b}} = -2\sqrt{b}
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can multiplying by the conjugate help simplify other limits involving square roots?
2. What are other types of indeterminate forms, and how can they be solved?
3. Can L'Hôpital's Rule be applied to this limit problem?
4. What is the difference of squares identity, and how is it used in calculus?
5. How does the limit change if \( b \) is negative or zero?

**Tip:** When dealing with square roots in limits, multiplying by the conjugate is often a helpful trick to simplify expressions and eliminate indeterminate forms.

Evaluate the limit: lim (x -> b) (b - x) / (sqrt(x) - sqrt(b))

Learn how to evaluate the limit lim (x -> b) (b - x) / (sqrt(x) - sqrt(b)) using algebraic manipulation and the conjugate method. This problem involves limits, indeterminate forms, and difference of squares.

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