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The problem gives us several limits of functions \( f(x) \) and \( g(x) \) as \( x \to 5 \):

\[
\lim_{x \to 5} f(x) = 4, \quad \lim_{x \to 5} f'(x) = 2, \quad \lim_{x \to 5} g(x) = 5
\]

We are asked to evaluate \( \lim_{x \to 5} (f'(x) + 9g(x)) \).

### Step-by-step solution:
1. Using the limit properties, we can separate the limits:
   \[
   \lim_{x \to 5} (f'(x) + 9g(x)) = \lim_{x \to 5} f'(x) + \lim_{x \to 5} 9g(x)
   \]

2. Now, substitute the given values:
   - \( \lim_{x \to 5} f'(x) = 2 \)
   - \( \lim_{x \to 5} g(x) = 5 \)

   Therefore:
   \[
   \lim_{x \to 5} (f'(x) + 9g(x)) = 2 + 9 \times 5
   \]

3. Perform the calculation:
   \[
   2 + 45 = 47
   \]

### Final answer:
\[
\lim_{x \to 5} (f'(x) + 9g(x)) = 47
\]

Would you like more details or have any further questions on this?

Here are 5 related questions:
1. How do you use limit properties for summation of functions?
2. What happens when the limit of a derivative is involved?
3. Can you generalize this approach for other combinations of functions?
4. How would you compute limits involving products of functions?
5. What does the derivative \( f'(x) \) tell us about the behavior of the function \( f(x) \)?

**Tip**: When solving limit problems involving derivatives, always check if the derivative values are directly given, as this simplifies the process.

The table above gives selected limits of the functions f and g. What is lim (f'(x) + 9g(x)) as x approaches 5?

Learn how to evaluate the limit of the expression f'(x) + 9g(x) as x approaches 5 using limit properties and given values for the functions f and g. Understand key concepts such as the limit sum rule and derivative limit evaluation, suitable for students in Grades 11-12.

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