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We are asked to find the following limit:

\[
\lim_{{x \to 2}} [5f(x) + g(x)]
\]

We are given:

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

To solve this, we can apply the limit laws, specifically the sum and scalar multiplication laws. The limit of a sum is the sum of the limits, and constants can be factored out of limits.

### Step-by-step:
1. First, apply the scalar multiplication rule:
   \[
   \lim_{{x \to 2}} 5f(x) = 5 \cdot \lim_{{x \to 2}} f(x) = 5 \cdot (-2) = -10
   \]
   
2. Now, apply the sum rule:
   \[
   \lim_{{x \to 2}} [5f(x) + g(x)] = \lim_{{x \to 2}} 5f(x) + \lim_{{x \to 2}} g(x)
   \]
   Substitute the limits:
   \[
   -10 + 4 = -6
   \]

Thus, the limit is:
\[
\lim_{{x \to 2}} [5f(x) + g(x)] = -6
\]

Would you like more details or have any questions about the solution?

Here are 5 related questions to expand your understanding:
1. What are the general rules for limits of sums and products?
2. How does scalar multiplication affect a limit?
3. Can a function's limit exist if the function itself is undefined at that point?
4. How do you handle limits when given piecewise functions?
5. What happens when both limits in a sum approach infinity?

**Tip:** When solving limit problems, always check if the basic limit laws (sum, product, scalar multiplication) can be applied before considering more advanced techniques.

Given that lim f(x) as x approaches 2 is -2 and lim g(x) as x approaches 2 is 4, find the following limit: lim [5f(x) + g(x)] as x approaches 2.

Learn how to find the limit of 5f(x) + g(x) as x approaches 2, given that lim f(x) = -2 and lim g(x) = 4. This problem involves applying limit laws, including the sum and scalar multiplication rules. Suitable for high school students.

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