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The problem asks to determine whether the function:

\[
f(x) = \frac{x^2 - 4}{x - 2}
\]

is continuous at \( x = 2 \), using the definition of continuity.

### Step-by-step solution:

1. **Simplify the function**:
   The expression \( x^2 - 4 \) can be factored as \( (x - 2)(x + 2) \), so the function becomes:
   \[
   f(x) = \frac{(x - 2)(x + 2)}{x - 2}
   \]
   For \( x \neq 2 \), we can cancel \( x - 2 \) from both the numerator and the denominator, leaving:
   \[
   f(x) = x + 2 \quad \text{for} \quad x \neq 2
   \]

2. **Definition of continuity**:
   A function \( f(x) \) is continuous at \( x = 2 \) if:
   \[
   \lim_{x \to 2} f(x) = f(2)
   \]
   This means that the limit of the function as \( x \) approaches 2 must equal the value of the function at \( x = 2 \).

3. **Evaluate the limit**:
   For \( x \neq 2 \), we have \( f(x) = x + 2 \). Now, take the limit as \( x \) approaches 2:
   \[
   \lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 2 + 2 = 4
   \]

4. **Value of the function at \( x = 2 \)**:
   The original function \( f(x) = \frac{x^2 - 4}{x - 2} \) is undefined at \( x = 2 \), since substituting \( x = 2 \) into the denominator gives zero. Therefore, \( f(2) \) is not defined.

5. **Conclusion**:
   Since \( f(2) \) is not defined, the function is **not continuous** at \( x = 2 \), even though the limit exists. The discontinuity is a removable discontinuity because the limit exists, but the function is not defined at that point.

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Would you like more details or have any questions?

Here are some related questions to explore:
1. What is the precise definition of a removable discontinuity?
2. How can you modify this function to make it continuous at \( x = 2 \)?
3. What is the difference between removable and non-removable discontinuities?
4. How can L'Hopital's Rule be applied to find limits of indeterminate forms like \( \frac{0}{0} \)?
5. What are other types of discontinuities besides removable?

**Tip**: When a function has a removable discontinuity, it can often be redefined to make it continuous by properly defining the function at that point.

Using the definition, determine whether the function f(x) = (x^2 - 4) / (x - 2) is continuous at x = 2 and justify the conclusion.

Explore whether the function f(x) = (x^2 - 4) / (x - 2) is continuous at x = 2 using the definition of continuity. This problem involves key concepts in calculus, such as limits, factoring, and removable discontinuities, suitable for Grades 11-12.

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