To evaluate the limit

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

we notice that as $x$ approaches infinity, both $\ln(x)$ and $4x$ approach infinity, resulting in an indeterminate form of type $\frac{\infty}{\infty}$. We can apply L'Hôpital's rule, which states that for limits of the form $\frac{\infty}{\infty}$, we can take the derivative of the numerator and the denominator.

Taking the derivatives:

1. The derivative of the numerator, $\ln(x)$, is $\frac{1}{x}$.
2. The derivative of the denominator, $4x$, is $4$.

Now, applying L'Hôpital's rule:

$\lim_{x \to \infty} \frac{\ln(x)}{4x} = \lim_{x \to \infty} \frac{\frac{1}{x}}{4} = \lim_{x \to \infty} \frac{1}{4x}.$

As $x$ approaches infinity, $\frac{1}{4x}$ approaches $0$:

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

Substituting back into the original limit we are evaluating:

$\lim_{x \to \infty} (4x - \ln(x)) = \lim_{x \to \infty} 4x - \lim_{x \to \infty} \ln(x) = \infty - 0 = \infty.$

Thus, the overall limit is:

$\lim_{x \to \infty} (4x - \ln(x)) = \infty.$

Summary:

1. The limit $\lim_{x \to \infty} \frac{\ln(x)}{4x} = 0$.
2. Therefore, $\lim_{x \to \infty} (4x - \ln(x)) = \infty$.

Would you like more details on any part of this process, or do you have other questions? Here are some related questions you might find interesting:

1. What are some other examples where L'Hôpital's rule can be applied?
2. How does the growth rate of logarithmic functions compare to polynomial functions?
3. Can you provide examples of limits that result in forms other than $\frac{\infty}{\infty}$?
4. What happens to the limit if we change the coefficients in $4x$ to a larger number?
5. How does the concept of limits apply in real-world scenarios?

Tip: When dealing with limits approaching infinity, always check if the form is indeterminate, as this often indicates the use of L'Hôpital's rule or algebraic manipulation.