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The given function is:

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

We need to find the values of \( x = a \) where the function is discontinuous, and for each value of \( a \), determine the limit of the function as \( x \) approaches \( a \). We also need to note if the limit does not exist.

### Step 1: Simplify the function
The function \( f(x) = \frac{x^2 - 4}{x - 2} \) contains a quadratic expression in the numerator. We can factor the numerator:

\[
x^2 - 4 = (x - 2)(x + 2)
\]

Thus, the function becomes:

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

For \( x \neq 2 \), we can cancel the \( (x - 2) \) terms, and the simplified function is:

\[
f(x) = x + 2 \quad \text{for} \quad x \neq 2
\]

### Step 2: Identify discontinuity
The function is simplified to \( f(x) = x + 2 \), but there is still a discontinuity at \( x = 2 \) because the original function had the factor \( \frac{0}{0} \) at \( x = 2 \). Therefore, \( f(x) \) is **discontinuous at \( x = 2 \)**.

### Step 3: Determine the limit as \( x \) approaches \( 2 \)
We now compute the limit of \( f(x) \) as \( x \) approaches \( 2 \). Since the simplified function is \( f(x) = x + 2 \) for \( x \neq 2 \), we can compute the limit as:

\[
\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 2 + 2 = 4
\]

Thus, the limit of \( f(x) \) as \( x \) approaches \( 2 \) is \( 4 \).

### Conclusion:
- The function is discontinuous at \( x = 2 \).
- The limit of the function as \( x \) approaches \( 2 \) is \( 4 \), but the function is not defined at \( x = 2 \) itself, causing the discontinuity.

Would you like more details or explanations about any specific steps?

### Related Questions:
1. What is the formal definition of a removable discontinuity?
2. Can we make the function continuous at \( x = 2 \)? How?
3. How would the behavior of the function change if the denominator was \( x - 3 \)?
4. What happens to the limit of the function as \( x \to \infty \)?
5. How would this problem change if there was an additional term like \( \frac{1}{x-2} \) added to the function?

**Tip:** Always check for factors that can be canceled to simplify rational functions before evaluating limits.

Find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn't exist. f(x) = (x^2 - 4) / (x - 2)

Learn how to find the points of discontinuity and evaluate the limit of the function f(x) = (x^2 - 4) / (x - 2). This solution explains the concept of removable discontinuity and how to factor quadratics to simplify rational functions. Suitable for Grades 10-12.

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