It appears you're working on a problem involving limits and derivatives, specifically focusing on differentiating a sum of functions. This setup resembles the limit definition of a derivative applied to a sum, which uses the property:

$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = f'(x)$

The missing expression at the end, labeled as "Blank 6," should likely represent the numerator in the limit definition of the derivative for $f(x)$, suggesting:

$f(x + h) - f(x)$

This matches the last option in your answer choices. Here's how it fits into the context:

The expression has been broken down by distributing the limit across the sum and applying it separately to each function $g(x)$ and $f(x)$, preparing it to show that the derivative of $g(x) + f(x)$ is the sum of $g'(x) + f'(x)$.

Would you like a detailed breakdown of each step in this problem?

Related Questions:

1. What is the limit definition of a derivative, and how is it used here?
2. How does the limit apply separately to the terms involving $g(x)$ and $f(x)$?
3. Why is $f(x+h) - f(x)$ the correct choice for "Blank 6"?
4. What is the derivative of a sum of functions in general?
5. How can the limit property be applied to sums and constants?

Tip:

When working with limits, remember that if the limit of each individual part exists, the limit of their sum is simply the sum of their limits.