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The image you uploaded contains a mathematical expression involving a series of limits and a question asking to determine the values of \( \alpha \) and \( \beta \) in a particular context.

To solve this problem, let's go through the steps and concepts used to determine the values of \( \alpha \) and \( \beta \).

1. **Given Expression:**
   \[
   \lim_{x \to 0} \frac{f(x)}{x} = \alpha
   \]
   \[
   \lim_{x \to 0} \frac{g(x)}{x} = \beta
   \]
   We are asked to evaluate the expression \( \lim_{x \to 0} \frac{f(x)g(x)}{x^2} \).

2. **Rewriting the Given Expression:**
   We can rewrite the limit expression for \( \lim_{x \to 0} \frac{f(x)g(x)}{x^2} \) as:
   \[
   \lim_{x \to 0} \frac{f(x)g(x)}{x^2} = \lim_{x \to 0} \left(\frac{f(x)}{x} \cdot \frac{g(x)}{x}\right)
   \]
   From the given limits:
   \[
   \lim_{x \to 0} \frac{f(x)}{x} = \alpha
   \]
   \[
   \lim_{x \to 0} \frac{g(x)}{x} = \beta
   \]
   Therefore:
   \[
   \lim_{x \to 0} \frac{f(x)g(x)}{x^2} = \alpha \cdot \beta
   \]

3. **Conclusion:**
   The limit \( \lim_{x \to 0} \frac{f(x)g(x)}{x^2} \) equals \( \alpha \cdot \beta \). Thus, the values of \( \alpha \) and \( \beta \) are simply multiplied together.

### Final Answer:
- The expression \( \lim_{x \to 0} \frac{f(x)g(x)}{x^2} \) simplifies to \( \alpha \cdot \beta \).

Would you like more details on any specific step, or do you have any other questions?

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Here are 5 related questions:
1. What are the interpretations of limits in calculus?
2. How do you compute the limit of a product of two functions?
3. What does it mean if a limit of a function as \( x \) approaches 0 is finite?
4. How would the result change if \( \lim_{x \to 0} \frac{f(x)}{x} = 0 \)?
5. What are the techniques for evaluating more complex limits involving trigonometric functions?

**Tip:** When dealing with limits, always check if you can simplify the expression before taking the limit to avoid unnecessary complications.

Learn how to solve the limit \( \lim_{x \to 0} \frac{f(x)g(x)}{x^2} \) using the product rule for limits. Understand the concepts of limits, including how to compute limits involving products of functions. Suitable for college-level students and those studying calculus.

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