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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.

Select the correct option for 'Blank 6' in the limit problem involving differentiation of a sum of functions.

Explore the limit definition of the derivative for differentiating a sum of functions. This problem applies the property of limits in calculus to show that the derivative of \( g(x) + f(x) \) is \( g'(x) + f'(x) \). Perfect for students in advanced high school or introductory college calculus courses.

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