To solve the given limit:

$\lim_{x \to \infty} \frac{4(\ln x)^3}{x}$

we can apply L'Hôpital's Rule because it is an indeterminate form of type $\frac{\infty}{\infty}$.

Step-by-Step Solution:

1. Apply L'Hôpital's Rule: L'Hôpital's Rule states that if the limit is in an indeterminate form, we can take the derivative of the numerator and the denominator:

   [ \lim_{x \to \infty} \frac{4(\ln x)^3}{x} = \lim_{x \to \infty} \frac{\frac{d}{dx} \left[ 4(\ln x)^3 \right]}{\frac{d}{dx} [x]} ]

2. Differentiate the numerator and denominator:

   • Derivative of the numerator:

     $\frac{d}{dx} \left[ 4(\ln x)^3 \right] = 4 \cdot 3(\ln x)^2 \cdot \frac{1}{x} = \frac{12(\ln x)^2}{x}$

   • Derivative of the denominator:

     $\frac{d}{dx} [x] = 1$

   So the limit becomes:

   $\lim_{x \to \infty} \frac{\frac{12(\ln x)^2}{x}}{1} = \lim_{x \to \infty} \frac{12(\ln x)^2}{x}$

3. Apply L'Hôpital's Rule again because the limit is still of the form $\frac{\infty}{\infty}$:

   [ \lim_{x \to \infty} \frac{12(\ln x)^2}{x} = \lim_{x \to \infty} \frac{\frac{d}{dx} \left[ 12(\ln x)^2 \right]}{\frac{d}{dx} [x]} ]

4. Differentiate the numerator again:

   • Derivative of $12(\ln x)^2$:

     $\frac{d}{dx} \left[ 12(\ln x)^2 \right] = 12 \cdot 2(\ln x) \cdot \frac{1}{x} = \frac{24 \ln x}{x}$

   So the limit becomes:

   $\lim_{x \to \infty} \frac{\frac{24 \ln x}{x}}{1} = \lim_{x \to \infty} \frac{24 \ln x}{x}$

5. Apply L'Hôpital's Rule one more time:

   [ \lim_{x \to \infty} \frac{24 \ln x}{x} = \lim_{x \to \infty} \frac{\frac{d}{dx} [24 \ln x]}{\frac{d}{dx} [x]} ]

   • Derivative of $24 \ln x$:

     $\frac{d}{dx} [24 \ln x] = \frac{24}{x}$

   So the limit is:

   $\lim_{x \to \infty} \frac{\frac{24}{x}}{1} = \lim_{x \to \infty} \frac{24}{x} = 0$

Final Answer:

$\lim_{x \to \infty} \frac{4(\ln x)^3}{x} = 0$

Would you like more details or have any questions?

Related Questions:

1. What is L'Hôpital's Rule, and when can it be applied?
2. How would the solution change if the numerator was $(\ln x)^4$ instead?
3. What are other indeterminate forms that L'Hôpital's Rule can address?
4. How does the growth rate of $\ln x$ compare to polynomials or exponential functions as $x \to \infty$?
5. Can the limit be solved using a method other than L'Hôpital’s Rule?

Tip:

Remember, L'Hôpital's Rule can only be applied to limits that are in indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.