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The problem provided is the following:

\[
f(x) = \frac{x^2 + 3x - 10}{x^2 - 4} = \frac{(x + 5)(x - 2)}{(x - 2)(x + 2)}
\]
You are asked to evaluate the limit:

\[
\lim_{x \to -2} f(x)
\]

### Step-by-Step Solution:

1. **Factor the numerator and the denominator:**
   \[
   f(x) = \frac{(x + 5)(x - 2)}{(x - 2)(x + 2)}
   \]
   
2. **Cancel the common factor \((x - 2)\):**
   For \(x \neq 2\), we can simplify the function:
   \[
   f(x) = \frac{x + 5}{x + 2}
   \]

3. **Evaluate the limit:**
   Now, we compute the limit as \(x\) approaches \(-2\):
   \[
   \lim_{x \to -2} \frac{x + 5}{x + 2}
   \]

4. **Substitute \(x = -2\) directly into the simplified expression:**
   \[
   \lim_{x \to -2} \frac{x + 5}{x + 2} = \frac{-2 + 5}{-2 + 2} = \frac{3}{0}
   \]

5. **Conclusion:**
   Since dividing by zero is undefined, the limit does not exist. However, because we get a positive number divided by zero, we need to analyze the behavior as \(x\) approaches \(-2\) from both sides:

   - As \(x \to -2^+\) (from the right), \(x + 2 > 0\), leading to \(+\infty\).
   - As \(x \to -2^-\) (from the left), \(x + 2 < 0\), leading to \(-\infty\).

   Thus, the limit as \(x\) approaches \(-2\) is undefined due to a vertical asymptote, but we observe different behaviors from either side.

### Final Answer:
\[
\lim_{x \to -2} f(x) = \text{undefined} \quad \text{(positive infinity from the right, negative infinity from the left)}.
\]

Would you like more details or have any questions?

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Here are 5 related questions you might consider:
1. What happens when you evaluate the limit at \(x = 2\)?
2. How can you determine if a function has a vertical asymptote at a given point?
3. What is the behavior of \(f(x)\) as \(x \to \infty\)?
4. How would the answer change if you approached a different type of indeterminate form (e.g., \(0/0\))?
5. Can this function be redefined to remove the discontinuity at \(x = 2\)?

**Tip:** When evaluating limits that result in \(\frac{c}{0}\), analyze the behavior of the function from both sides to determine if there is a vertical asymptote or if the limit does not exist.

lim(x→-2) f(x), where f(x) = (x^2 + 3x - 10) / (x^2 - 4)

Learn how to evaluate the limit of the rational function f(x) = (x^2 + 3x - 10) / (x^2 - 4) as x approaches -2. This solution demonstrates the use of factorization, simplifying the expression, and analyzing vertical asymptotes. Suitable for students in Grades 11-12.

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