The problem asks to find the limit $\lim_{x \to 0} g(x)$, given the following information:

1. $\lim_{x \to 0} f(x) = 0$
2. $\lim_{x \to 0} (3f(x) + 3g(x)) = 7$

Step-by-Step Solution:

We can start with the second condition:

$\lim_{x \to 0} (3f(x) + 3g(x)) = 7$

Since limits distribute over addition and scalar multiplication, we can split this limit into two parts:

$3 \lim_{x \to 0} f(x) + 3 \lim_{x \to 0} g(x) = 7$

Using the given information that $\lim_{x \to 0} f(x) = 0$, this simplifies to:

$3(0) + 3 \lim_{x \to 0} g(x) = 7$

Which reduces to:

$3 \lim_{x \to 0} g(x) = 7$

Now, solve for $\lim_{x \to 0} g(x)$:

$\lim_{x \to 0} g(x) = \frac{7}{3}$

Final Answer:

$\lim_{x \to 0} g(x) = \frac{7}{3}$

Would you like more details or have any further questions?

Here are 5 related questions to expand on this:

1. How would the answer change if $\lim_{x \to 0} f(x)$ were not zero?
2. Can we use L'Hopital's Rule to solve similar limit problems?
3. What happens if $\lim_{x \to 0} (3f(x) + 3g(x))$ were undefined?
4. How do we handle situations when $\lim_{x \to 0} g(x)$ does not exist?
5. How can we verify if a given function has a limit using graphical methods?

Tip: Always apply the property of limits carefully when splitting terms, especially when dealing with functions that involve addition and multiplication.