Question

How do you find the Limit of $\dfrac{{\ln (n)}}{n}$ as $n$ approaches infinity?

Answer 1

Hint : The first thing to try and do is to actually understand after you should use L'Hôpital's Rule. L'Hôpital's Rule is a brilliant technique for addressing the limits of an indeterminate form.
An indeterminate form is when the limit seems to approach into a deeply weird answer.

Complete step by step solution:
L'Hôpital's Rule says:
If you have got an indeterminate form for your answer to your limit, then you'll take the derivative of the numerator and of the denominator separately so as to search out the limit.
You can repeat this process if you still get an indeterminate form. An indeterminate form is when the limit seems to approach into a deeply weird answer.
The given limit is in the indeterminate form $\dfrac{\infty }{\infty }$ , so we can apply L'Hôpital's rule,
$\mathop {\lim }\limits_{n \to \infty } \dfrac{{\ln (n)}}{n}$
Now, taking the derivative of the numerator and of the denominator separately so that we can find the limit. Therefore, we will get the following result,
$= \mathop {\lim }\limits_{n \to \infty } \dfrac{{\dfrac{1}{n}}}{1}$
Now, since I've taken the derivative of the numerator and therefore the denominator separately.
Now, substitute $n = \infty$ , we will get,
$= \mathop {\lim }\limits_{n \to \infty } \dfrac{1}{n} = \dfrac{1}{\infty } = 0$
If I had found the solution to still in indeterminate form, then I might need to continue using L'Hôpital's Rule.
So, the correct answer is “0”.

Note : If I had found the solution to still in indeterminate form, then I might need to continue using this L'Hôpital's Rule. But as soon as I get a determinate form , no need to do further. Because our solution is no longer in an indeterminate form, therefore L'Hôpital's Rule does not apply.