Answer
$\ln3$.
Work Step by Step
Since we have
$$\lim_{n\to \infty}a_n=\lim_{n\to \infty} \ln\left(\frac{12n+2}{-9+4n}\right)
=\lim_{n\to \infty}\ln\left(\frac{12+(2/n)}{-(9/n)+4}\right)\\
=\ln\left(\frac{12+0}{0+4}\right)=\ln3,$$
then the sequence $a_n$ converges to $\ln3$.
