Question

How do you find the Limit of `ln (n)/ n` as n approaches infinity?

Answer 1

`0`

Explanation:

This is in the indeterminate form `oo/oo`, so we can apply l'Hôpital's rule, which states that we can take the derivative of the numerator and denominator and then plug in `oo` again to find the limit. Therefore

`lim_(nrarroo)ln(n)/n=lim_(nrarroo)(1/n)/1=lim_(nrarroo)1/n=1/oo=0`

We can also analyze this intuitively: the linear function `n` rises at a greater rate than the logarithmic function `ln(n)`, so since the function that rises faster is in the denominator, the function will approach `0`.

If the function had been flipped, we'd see that the limit as `n` approaches `oo` in `n/ln(n)` would be `oo`.