_ n n(2n+1)2 (n+2)(n2+3n−1) is equal to: - Vidyadip

MCQ

$\lim_\limits{\text{n} \rightarrow \infty}\frac{\text{n}(2\text{n}+1)2}{(\text{n}+2)(\text{n}2+3\text{n}−1)}$ is equal to:

A
$0$
B
$2$
✓
$4$
D
$ \infty$

✓

Answer

Correct option: C.

$4$

$=\lim_\limits{\text{n} \rightarrow \infty}\frac{\text{n}(2\text{n}+1)2}{(\text{n}+2)(\text{n}2+3\text{n}−1)}$
$ = \displaystyle \lim_{\text{n}\to\infty}{\displaystyle \frac {\left(2+\Large \frac{1}{\text{n}} \right)^2}{\left(1+\Large \frac{2}{\text{n}} \right)\left(1+\Large \frac{3}{\text{n}} - \Large \frac{1}{\text{n}^2} \right)} }$
$=\text{n}$
$ = \displaystyle \frac{(2+0)^2}{(1+0)(1+0+0)}$

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