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The limit expression given is:

\[
\lim_{x \to 4} \sqrt{x^2 - 2x - 4}
\]

### Step-by-Step Solution:

1. **Check if the function is continuous**:
   - The function involves a square root, so it's important to check that the expression inside the square root, \( x^2 - 2x - 4 \), is non-negative at and near \( x = 4 \), because the square root function is only defined for non-negative values in the reals.
   
2. **Substitute \( x = 4 \) directly into the function**:
   Since the limit involves a continuous function (square root), we can try direct substitution if the expression inside the square root is non-negative.

   First, evaluate \( x^2 - 2x - 4 \) at \( x = 4 \):

   \[
   4^2 - 2(4) - 4 = 16 - 8 - 4 = 4
   \]

   Thus, the expression inside the square root becomes:

   \[
   \sqrt{4} = 2
   \]

3. **Conclusion**:
   Since substituting \( x = 4 \) into the expression gives a real and finite result, the limit is:

   \[
   \boxed{2}
   \]

Would you like further clarification or details on any of these steps?

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Here are five related questions to deepen your understanding:

1. How do you determine if a function involving a square root is continuous?
2. What would happen if the expression inside the square root were negative at \( x = 4 \)?
3. How would you approach solving this limit if direct substitution didn't work?
4. What is the general form of a continuous function, and why can we directly substitute in this case?
5. How can you apply the limit laws to solve more complex limits involving square roots?

**Tip**: Always check the domain of functions involving square roots or logarithms before applying limit rules directly!

Learn how to evaluate the limit of the function involving a square root as x approaches 4 using continuity. This detailed step-by-step solution covers key calculus concepts including limits, continuity, and square roots, suitable for Grades 11-12.

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