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The limit problem you provided is:

\[
\lim_{{x \to 16}} \frac{\sqrt{x} - 4}{x - 16}
\]

### Step-by-step Solution:
1. **Substitute \( x = 16 \):**
   \[
   \frac{\sqrt{16} - 4}{16 - 16} = \frac{4 - 4}{0} = \frac{0}{0}
   \]
   This results in an indeterminate form \( \frac{0}{0} \), so we need to manipulate the expression.

2. **Multiply by the conjugate:**
   To resolve the indeterminate form, multiply the numerator and denominator by the conjugate of the numerator:
   \[
   \frac{\sqrt{x} - 4}{x - 16} \times \frac{\sqrt{x} + 4}{\sqrt{x} + 4} = \frac{(\sqrt{x} - 4)(\sqrt{x} + 4)}{(x - 16)(\sqrt{x} + 4)}
   \]

3. **Simplify the numerator:**
   The numerator becomes a difference of squares:
   \[
   (\sqrt{x})^2 - 4^2 = x - 16
   \]
   So, the expression simplifies to:
   \[
   \frac{x - 16}{(x - 16)(\sqrt{x} + 4)} = \frac{1}{\sqrt{x} + 4}
   \]
   Now, we can substitute \( x = 16 \) into this simplified expression.

4. **Substitute \( x = 16 \):**
   \[
   \frac{1}{\sqrt{16} + 4} = \frac{1}{4 + 4} = \frac{1}{8}
   \]

Thus, the limit is:

\[
\boxed{\frac{1}{8}}
\]

### Do you want more details or have any questions?

Here are 5 related questions to explore:
1. What is a conjugate, and why does multiplying by it help resolve indeterminate forms?
2. How can L'Hopital's Rule be used to solve indeterminate forms like \( \frac{0}{0} \)?
3. What are other common techniques for solving limits involving square roots?
4. How does the limit change if the expression inside the square root is more complex?
5. What happens if we approach a different value of \( x \) other than 16?

**Tip:** When faced with indeterminate forms like \( \frac{0}{0} \), always consider algebraic simplifications such as multiplying by conjugates or factoring!

lim (sqrt(x) - 4) / (x - 16) as x approaches 16

Solve the limit problem lim (sqrt(x) - 4) / (x - 16) as x approaches 16 using conjugate multiplication to handle the indeterminate form 0/0. This step-by-step guide explains key limit concepts and theorems, ideal for high school students (Grades 11-12).

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