Let's analyze the given equations and determine which one is correct.

The problem appears to involve the definition of a derivative:

$g'(z) = \lim_{h \to 0} \frac{g(z+h) - g(z)}{h}$

We examine the choices:

• Option A simplifies incorrectly because the numerator should not have a "+4".
• Option B has an incorrect numerator.
• Option C appears to have the wrong denominator structure.
• Option D correctly simplifies the numerator and denominator.

Thus, option D is the correct answer.

Would you like a step-by-step breakdown of why option D is correct? Here are five related questions to explore:

1. How do you formally apply the definition of a derivative to rational functions?
2. What are common mistakes when simplifying difference quotients?
3. How does factoring help in evaluating derivative limits?
4. What happens when the denominator does not cancel properly in a derivative computation?
5. Can this approach be extended to higher-order derivatives?

Tip:

Always ensure that the numerator correctly reflects $g(z+h) - g(z)$ before simplifying!