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The given expression is:

\[
\lim_{x \to 36} \frac{x - 36}{\sqrt{x} - 6}
\]

### Step-by-Step Solution:

1. **Direct Substitution**:
   - First, attempt to substitute \( x = 36 \) directly into the expression:
   \[
   \frac{36 - 36}{\sqrt{36} - 6} = \frac{0}{6 - 6} = \frac{0}{0}
   \]
   This results in the indeterminate form \( \frac{0}{0} \), so we need to apply a different method, such as **factoring** or **rationalizing** the denominator.

2. **Rationalizing the Denominator**:
   - To eliminate the square root in the denominator, multiply both the numerator and denominator by the conjugate of the denominator:
   \[
   \frac{x - 36}{\sqrt{x} - 6} \times \frac{\sqrt{x} + 6}{\sqrt{x} + 6} = \frac{(x - 36)(\sqrt{x} + 6)}{(\sqrt{x} - 6)(\sqrt{x} + 6)}
   \]
   The denominator simplifies using the difference of squares:
   \[
   (\sqrt{x} - 6)(\sqrt{x} + 6) = x - 36
   \]
   Now the expression becomes:
   \[
   \frac{(x - 36)(\sqrt{x} + 6)}{x - 36}
   \]
   Since \( x - 36 \) appears in both the numerator and denominator, we can cancel them out (as long as \( x \neq 36 \)):
   \[
   \lim_{x \to 36} (\sqrt{x} + 6)
   \]

3. **Final Step**:
   - Now, substitute \( x = 36 \) into the simplified expression:
   \[
   \sqrt{36} + 6 = 6 + 6 = 12
   \]

Thus, the limit is:

\[
\boxed{12}
\]

Would you like further details or have any questions? Here are 5 related questions to deepen your understanding:

1. How would you approach a limit problem involving a square root and a polynomial?
2. Can you explain the difference of squares method used in rationalizing the denominator?
3. How would you handle a limit problem where direct substitution results in a different indeterminate form?
4. What are other methods to solve limits when the indeterminate form \( \frac{0}{0} \) appears?
5. Why is it important to verify that \( x \neq 36 \) after canceling terms?

**Tip:** When solving limits involving square roots, rationalizing the denominator is often a helpful technique to eliminate indeterminate forms.

Find the limit: lim(x→36) (x - 36) / (√x - 6)

Learn how to solve the limit problem lim(x→36) (x - 36) / (√x - 6) using rationalization techniques and algebraic simplification. This solution includes key steps and explanations of the limit theorem and rationalizing square roots.

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