To find the derivative of the function $f(x) = 4 \ln(x + 5)$ using the limit definition of a derivative, we start with the standard definition of the derivative of a function $f(x)$:

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

For the given function $f(x) = 4 \ln(x + 5)$, let's plug this into the definition:

$f'(x) = \lim_{h \to 0} \frac{4 \ln((x+h) + 5) - 4 \ln(x + 5)}{h}$

We can factor out the constant 4 from the numerator:

$f'(x) = 4 \lim_{h \to 0} \frac{\ln((x+h) + 5) - \ln(x + 5)}{h}$

This is the expression for the derivative of the function $f(x)$ using the limit definition. There’s no need to simplify further as per the problem’s instructions.

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Would you like to explore this limit further or have any questions about the process? Here are some related questions to consider:

1. How does the derivative of a natural logarithmic function behave in general?
2. What is the next step if we wanted to simplify this expression?
3. How would you evaluate this derivative using rules of logarithmic differentiation?
4. How would you use this approach to find the derivative of a different logarithmic function?
5. Can you compare this limit form to the chain rule method of differentiation?

Tip: When differentiating logarithmic functions, remember that the derivative of $\ln(x)$ is $\frac{1}{x}$, which can guide simplifications.