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.