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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.

Let f and g be two functions and suppose that lim(x→0)f(x) = 0 and lim(x→0)(3f(x) + 3g(x)) = 7. If possible, find the value of lim(x→0)g(x) and enter your answer as an exact value.

Solve a calculus problem involving limits of functions f(x) and g(x), where lim(x→0)f(x) = 0 and lim(x→0)(3f(x) + 3g(x)) = 7. Learn how to apply limit properties to find lim(x→0)g(x).

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