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?

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.