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.