(1 point)
Find the limit. Use I'Hospital's Rule where appropriate.
\[
\lim _{x \rightarrow \infty} \frac{4(\ln x)^{2}}{x}
\]

Limit:

Step 1 ：The given limit is of the form \(\frac{\infty}{\infty}\), which is an indeterminate form. Therefore, we can apply L'Hopital's Rule. L'Hopital's Rule states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives, provided the original limit is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

Step 2 ：We need to find the derivative of the numerator and the derivative of the denominator, and then find the limit of the quotient of these two derivatives as x approaches infinity.

Step 3 ：The derivative of the numerator \(4\ln(x)^{2}\) is \(\frac{8\ln(x)}{x}\).

Step 4 ：The derivative of the denominator x is 1.

Step 5 ：The limit of the quotient of the derivatives as x approaches infinity is 0.

Step 6 ：Therefore, by L'Hopital's Rule, the limit of the original function as x approaches infinity is also 0.

Step 7 ：Final Answer: The limit is \(\boxed{0}\).