Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
$$a_n= \frac{\ln(4n)}{\ln(12n)},\ n\in\Bbb{N}$$
my first guess was to write in under one natural log as $\ln\left(\frac{4n}{2n}\right)$. since they have the same variable $n$, it can nicely reduce to $\ln(1/3)$. so the limit as n goes to infinity would be $\ln(1/3)$.
Why is this thinking wrong? When I have a rational function, if there are same variable raised to the same power, the limit would just approach that number. What makes this problem any different? why must I use L'Hospital for this problem
Hint:$$a_n=\frac{\ln(4n)}{\ln(12n)}=\frac{\ln(4n)}{\ln(3)+\ln(4n)}$$