_ n → ∞ (3)/(n) \ 4 + (2 + (1)/(n)) ^ 2 + (2 + (2)/(n)) ^ 2 + + (3...

Let

$L = \lim_{n \to \infty}\dfrac{3}{n}\left[4 + {\left(2 + \dfrac{1}{n}\right)}^{2} + {\left(2 + \dfrac{2}{n}\right)}^{2} + . . . + {\left(3 - \dfrac{1}{n}\right)}^{2}\right]$

$⇒ L = \lim_{n \to \infty}\dfrac{3}{n}\left[{\left(2 + \dfrac{0}{n}\right)}^{2} + {\left(2 + \dfrac{1}{n}\right)}^{2} + {\left(2 + \dfrac{2}{n}\right)}^{2} + . . . + {\left(2 + \left(1 - \dfrac{1}{n}\right)\right)}^{2}\right]$

$⇒ L = 3\lim_{n \to \infty}\dfrac{1}{n}\left[\mathop{∑}_{r = 0}^{n - 1}{\left(2 + \dfrac{r}{n}\right)}^{2}\right]$

$⇒ L = 3{∫}_{0}^{1}{\left(2 + x\right)}^{2}dx$

$⇒ L = {\left[{\left(2 + x\right)}^{3}\right]}_{0}^{1}$

$⇒ L = 27 - 8$

$⇒ L = 19$