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The limit problem you provided involves evaluating:

\[
\lim_{{x \to 2}} \frac{3f(x)}{g(x)}
\]

Given that:
- \(\lim_{{x \to 2}} f(x) = 4\)
- \(\lim_{{x \to 2}} g(x) = -2\)
- \(\lim_{{x \to 2}} h(x) = 0\)

We can proceed by applying the limit laws. Since both \(f(x)\) and \(g(x)\) have defined limits as \(x \to 2\), the limit of the quotient can be computed as:

\[
\lim_{{x \to 2}} \frac{3f(x)}{g(x)} = \frac{3 \cdot \lim_{{x \to 2}} f(x)}{\lim_{{x \to 2}} g(x)}
\]

Substituting the given limits:

\[
= \frac{3 \cdot 4}{-2} = \frac{12}{-2} = -6
\]

So the solution is:

\[
\lim_{{x \to 2}} \frac{3f(x)}{g(x)} = -6
\]

Would you like more details or have any additional questions?

Here are some related questions for deeper exploration:
1. What are the properties of limits that allow for this simplification?
2. How would the solution change if \(g(x)\) approached zero as \(x \to 2\)?
3. What happens if either \(f(x)\) or \(g(x)\) has an undefined limit at \(x = 2\)?
4. How would you apply L'Hopital's rule to similar limit problems?
5. Can you provide an example where limits involve piecewise functions?

**Tip:** Always check the denominator in a limit involving a quotient to avoid division by zero errors.

lim(x→2) [3f(x) / g(x)] with given that lim(x→2) f(x) = 4, lim(x→2) g(x) = -2, and lim(x→2) h(x) = 0

Solve the limit problem lim(x→2) 3f(x)/g(x) using limit laws. Given that lim(x→2) f(x) = 4, lim(x→2) g(x) = -2, apply the limit laws to find the solution. This example covers important calculus and algebra concepts suitable for high school students in Grades 11-12.

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