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To solve the given problem, we are asked to find the limit:

\[
\lim_{x \to 4} \left[3f(x) - 2g(x)\right]
\]

given that:

\[
\lim_{x \to 4} f(x) = 2 \quad \text{and} \quad \lim_{x \to 4} g(x) = -5.
\]

### Solution:

We can use the limit properties, specifically linearity, which states that:

\[
\lim_{x \to a} [af(x) + bg(x)] = a \lim_{x \to a} f(x) + b \lim_{x \to a} g(x).
\]

Applying this to our limit, we get:

\[
\lim_{x \to 4} \left[3f(x) - 2g(x)\right] = 3 \lim_{x \to 4} f(x) - 2 \lim_{x \to 4} g(x).
\]

Substituting the given values:

\[
= 3 \cdot 2 - 2 \cdot (-5).
\]

Calculating this:

\[
= 6 + 10 = 16.
\]

### Final Answer:

\[
\lim_{x \to 4} \left[3f(x) - 2g(x)\right] = 16.
\]

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. How does the linearity property of limits help simplify complex limit expressions?
2. What would the limit be if the coefficients of \(f(x)\) and \(g(x)\) were different?
3. How do you handle limits involving products and quotients?
4. Can you extend these limit rules to functions with higher complexity, such as compositions?
5. What happens if the given limits do not exist or are infinite?

**Tip:** Understanding and applying the properties of limits can simplify complex expressions, making them much easier to evaluate directly!

Suppose lim f(x) as x -> 4 = 2 and lim g(x) as x -> 4 = -5, find the lim as x -> 4 of [3f(x) - 2g(x)].

Solve the limit problem involving two functions f(x) and g(x) with given limits. Learn how to apply the linearity of limits to find the value of lim as x -> 4 of [3f(x) - 2g(x)]. This problem is suitable for students in Grades 10-12.

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