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The problem you've uploaded is asking to **apply L'Hopital's Rule** to evaluate the limit:

\[
\lim_{x \to 0} \frac{f(4 + 8x) + f(4 + 9x)}{x}
\]

### Steps to Apply L'Hopital's Rule

1. **Check the form**: To apply L'Hopital's Rule, the limit should be of the indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).
    - At \(x = 0\), the numerator becomes \(f(4) + f(4)\), and the denominator is \(0\). Assuming \(f(4) = 0\), this results in a \(\frac{0}{0}\) form, allowing us to apply L'Hopital's Rule.

2. **Differentiate the numerator and denominator**: L'Hopital's Rule tells us to differentiate both the numerator and denominator separately.

   - The derivative of the numerator \(f(4 + 8x) + f(4 + 9x)\) is:
     \[
     8f'(4 + 8x) + 9f'(4 + 9x)
     \]
   - The derivative of the denominator \(x\) is:
     \[
     1
     \]

3. **Rewrite the limit**:
   After applying L'Hopital's Rule, the limit becomes:
   \[
   \lim_{x \to 0} \left( 8f'(4 + 8x) + 9f'(4 + 9x) \right)
   \]

4. **Evaluate the limit as \(x \to 0\)**:
   - When \(x \to 0\), we substitute \(x = 0\) into the derivatives:
   \[
   8f'(4) + 9f'(4)
   \]
   This simplifies to:
   \[
   (8 + 9)f'(4) = 17f'(4)
   \]

### Final Answer:
\[
\boxed{17f'(4)}
\]

This is the result after applying L'Hopital's Rule to the given limit.

Let me know if you want more details or have any questions!

### Related Questions:
1. How can we confirm whether L'Hopital's Rule applies to a given limit?
2. What are other indeterminate forms where L'Hopital’s Rule can be applied?
3. How do we interpret the derivative \(f'(x)\) geometrically?
4. What happens if \(f'(4) = 0\) in this case?
5. Are there alternative methods to solve this limit without using L'Hopital's Rule?

**Tip**: Always check the form of a limit first before applying L'Hopital's Rule. If it's not an indeterminate form, L'Hopital’s Rule might not be applicable.

Apply L'Hopital's Rule to evaluate the limit lim(x→0) [f(4 + 8x) + f(4 + 9x)] / x.

Learn how to apply L'Hopital's Rule to evaluate the limit of (f(4 + 8x) + f(4 + 9x)) / x as x approaches 0. This problem involves advanced calculus concepts such as differentiation and the application of L'Hopital's Rule. Ideal for college calculus students.

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