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?

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.