\frac{ln(x)}{x} does in fact go to \frac{\infty}{\infty} when x tend to inf.
this is an indeterminate form. u can use l'hopital's rule to solve this easily.
easier way of doing this can be explained usingthe following example
lim x -->inf (x^5/x^4). lim x--> inf (x^5) is inf and lim x-->inf...