Answer
convergent with limit $3$
Work Step by Step
Given: $f(x)=\lim\limits_{n\to \infty} \dfrac{12}{n^2}[\dfrac{(n+1)(2n+1)]}{2}]^2= \lim\limits_{n\to \infty} \dfrac{3n^4+6n^3+3n^2}{n^4}$
The sequence converges when the limit $\lim\limits_{n\to \infty} a_n$ exists and when the limit $\lim\limits_{n\to \infty} a_n$ does not exist, then sequence diverges.
Here, we have
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty} \dfrac{3+(6/n)+(3/n^2)}{1}$
This gives:
$\lim\limits_{n\to \infty}a_n=\lim\limits_{n\to \infty}(3)+\lim\limits_{n\to \infty}\dfrac{6}{n}+\lim\limits_{n\to \infty}\dfrac{3}{n^2}$
Thus, $a_n=3$
Hence, the sequence is convergent with limit $3$
