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give me the correct answer and make it clear which answer is supposed to go in which blank

Let's go through the solution provided in the image step-by-step to determine the correct answers for each blank.

### Solution Walkthrough:

1. **Original Limit Expression:**
   \[
   \lim_{x \to \infty} \frac{\ln(4x)}{\sqrt[4]{x}}
   \]

2. **Identifying the Indeterminate Form:**
   As \( x \to \infty \), \( \ln(4x) \to \infty \) and \( \sqrt[4]{x} \to \infty \), which creates an indeterminate form \( \frac{\infty}{\infty} \). This suggests that we can apply **L'Hospital's Rule**.

3. **Applying L'Hospital's Rule:**
   Differentiate the numerator and denominator with respect to \( x \):
   \[
   \lim_{x \to \infty} \frac{\ln(4x)}{\sqrt[4]{x}} = \lim_{x \to \infty} \frac{\frac{1}{x}}{\frac{1}{4}x^{-3/4}}
   \]

4. **Simplifying the Result:**
   The expression simplifies further to:
   \[
   \lim_{x \to \infty} \frac{1}{x} \cdot \frac{4}{x^{-3/4}} = \lim_{x \to \infty} \frac{4}{x^{1/4}} 
   \]

5. **Evaluating the Final Limit:**
   As \( x \to \infty \), \( \frac{4}{x^{1/4}} \to 0 \).

### Filling in the Blanks:

- The first blank (next to "Since \(\ln(4x) \to \infty\) and") should be **"\(\sqrt[4]{x} \to \infty\)"**.
- The second blank at the end of the solution should be **"0"**.

### Final Answer:

1. **First Blank:** \(\sqrt[4]{x} \to \infty\)
2. **Second Blank:** \(0\)

Would you like a deeper explanation of any specific part of the solution?

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Here are 5 additional questions that build on this concept:

1. What are other examples of indeterminate forms that allow the use of L'Hospital's Rule?
2. Can we apply L'Hospital's Rule to limits that tend to \(0 \cdot \infty\)?
3. How would the solution change if the denominator was \(\sqrt{x}\) instead of \(\sqrt[4]{x}\)?
4. How do we know when L'Hospital’s Rule needs to be applied multiple times?
5. Why does \( \ln(4x) \) grow slower than any polynomial of \( x \)?

**Tip:** When using L'Hospital's Rule, always check if the limit after differentiation still leads to an indeterminate form; if not, further applications may be unnecessary.

Step-by-step solution to calculating the limit of lim x → ∞ ln(4x) / 4√x using L'Hospital's Rule. This problem covers advanced calculus concepts including limits, indeterminate forms, and differentiating logarithmic and root functions. Ideal for undergraduate calculus students.

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