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The image you've uploaded contains two limit expressions, which seem to involve two functions, \( f(x) \) and \( g(x) \), as \( x \to 0 \). Let me transcribe and analyze them:

1. The third equation is:

\[
\lim_{x \to 0} f(x)g(x)
\]

This represents the limit of the product of two functions, \( f(x) \) and \( g(x) \), as \( x \) approaches 0. To solve this, you'd need to know more about the behavior of \( f(x) \) and \( g(x) \) individually. If both limits exist, then:

\[
\lim_{x \to 0} f(x)g(x) = \lim_{x \to 0} f(x) \cdot \lim_{x \to 0} g(x)
\]

2. The fourth equation is:

\[
\lim_{x \to 0} \frac{f(x)}{g(x)}
\]

This represents the limit of the quotient \( \frac{f(x)}{g(x)} \) as \( x \to 0 \). Here, we need to examine whether the limit of \( g(x) \) approaches 0, and if so, whether we have an indeterminate form (like \( \frac{0}{0} \)). In such cases, L'Hôpital's Rule or algebraic simplification might be used to resolve the limit.

Do you have any specific functions for \( f(x) \) and \( g(x) \), or would you like further details on solving limits in general?

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Here are 5 related questions that could expand on this topic:

1. What happens if both \( f(x) \) and \( g(x) \) approach zero as \( x \to 0 \)?
2. How does L'Hôpital's Rule help when we encounter an indeterminate form in limits?
3. What are some common techniques for solving product limits like \( \lim_{x \to 0} f(x)g(x) \)?
4. Can you provide examples of functions \( f(x) \) and \( g(x) \) where the limit of the quotient leads to a finite result?
5. How do you handle situations where one function approaches infinity while the other approaches zero?

**Tip:** Always check if the limit results in indeterminate forms like \( \frac{0}{0} \) or \( \infty/\infty \), as these often require additional techniques such as L'Hôpital's Rule or factoring to solve.

The image uploaded by the user contains two limit expressions involving f(x) and g(x) as x approaches 0.

Explore how to solve limit problems involving the product and quotient of functions, f(x) and g(x), as x approaches 0. This guide covers important concepts such as indeterminate forms, L'Hôpital's Rule, and key theorems for handling limits of products and quotients. Suitable for advanced high school or early university students.

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