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Let's break this step-by-step and explain the derivation:

### Step 1: The given function \( f(x) \)
The function is defined as:
\[
f(x) = (x - 4) \sqrt[3]{x + 3}
\]

To evaluate \( f(5) \), substitute \( x = 5 \):
\[
f(5) = (5 - 4) \sqrt[3]{5 + 3}
\]
This simplifies to:
\[
f(5) = 1 \cdot \sqrt[3]{8} = 1 \cdot 2 = 2
\]

### Step 2: The derivative using limits
The derivative \( f'(x) \) at \( x = 5 \) is defined as:
\[
f'(5) = \lim_{h \to 0} \frac{f(5+h) - f(5)}{h}
\]

Since \( f(5) = 2 \), the numerator becomes \( f(5+h) - 2 \). Hence:
\[
f'(5) = \lim_{h \to 0} \frac{f(5+h) - 2}{h}
\]

### Step 3: Rewriting the limit
The solution explains splitting \( \frac{f(5+h) - 1}{h} \) into two parts:
\[
\frac{f(5+h) - 1}{5-h} \times \lim_{h \to 0} \frac{5-h - 2}{h}
\]

#### Explanation:
1. The substitution and factorization:
   - They compute \( f(5+h) - 2 \), showing how the derivative calculation relates to the limit.
   - \( f(x) \) is simplified in pieces for \( h \to 0 \).

Would you like me to expand or clarify any specific portion?

Please explain the steps in the provided solution for the given function and limit calculation.

Explore the detailed solution to finding the derivative of the function f(x) = (x - 4) * cube_root(x + 3) using the limit definition. This explanation covers limit rewriting and step-by-step simplifications. Suitable for high school calculus students.

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