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The given limit problem is:

\[
\lim_{{x \to 9}} \frac{\sqrt{7x - 62} - 1}{x - 9}
\]

### Approach:
This is an indeterminate form of type \( \frac{0}{0} \) when you directly substitute \( x = 9 \). To resolve this, we can use **rationalization** by multiplying both the numerator and the denominator by the conjugate of the numerator.

#### Step-by-step solution:

1. **Multiply by the conjugate:**
   Multiply both the numerator and the denominator by the conjugate of the numerator, \( \sqrt{7x - 62} + 1 \):

   \[
   \frac{\sqrt{7x - 62} - 1}{x - 9} \cdot \frac{\sqrt{7x - 62} + 1}{\sqrt{7x - 62} + 1} = \frac{(\sqrt{7x - 62})^2 - 1^2}{(x - 9)(\sqrt{7x - 62} + 1)}
   \]

2. **Simplify the numerator:**
   Use the difference of squares in the numerator:

   \[
   (\sqrt{7x - 62})^2 - 1^2 = (7x - 62) - 1 = 7x - 63
   \]

   So the expression becomes:

   \[
   \frac{7x - 63}{(x - 9)(\sqrt{7x - 62} + 1)}
   \]

3. **Factor the numerator:**
   Factor out \( 7 \) from the numerator:

   \[
   7(x - 9)
   \]

   Now the expression is:

   \[
   \frac{7(x - 9)}{(x - 9)(\sqrt{7x - 62} + 1)}
   \]

4. **Cancel out \( (x - 9) \):**

   Since \( x \neq 9 \), we can cancel \( (x - 9) \) from both the numerator and the denominator:

   \[
   \frac{7}{\sqrt{7x - 62} + 1}
   \]

5. **Substitute \( x = 9 \):**

   Now substitute \( x = 9 \) into the simplified expression:

   \[
   \frac{7}{\sqrt{7(9) - 62} + 1} = \frac{7}{\sqrt{63 - 62} + 1} = \frac{7}{\sqrt{1} + 1} = \frac{7}{1 + 1} = \frac{7}{2}
   \]

### Final Answer:

\[
\lim_{{x \to 9}} \frac{\sqrt{7x - 62} - 1}{x - 9} = \frac{7}{2}
\]

Would you like further details or have any other questions?

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Here are 5 related questions:
1. How do you recognize the indeterminate form of a limit problem?
2. What are some other methods for solving limits besides rationalization?
3. What happens if you don't cancel common factors in a limit problem?
4. How does the conjugate method help in limit problems with square roots?
5. How can you verify a limit using numerical methods?

**Tip:** Always check for factorable expressions or common terms in the numerator and denominator to simplify limits easily.

Evaluate the limit: lim(x→9) ((√(7x - 62) - 1) / (x - 9))

Learn how to evaluate the limit of (√(7x - 62) - 1) / (x - 9) as x approaches 9 using rationalization and the conjugate method to simplify the expression. Suitable for Grades 11-12, this problem demonstrates concepts in limits and indeterminate forms.

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