_ n n (2n + 1) ^2 (n + 2)( n^2 + 3n - 1) = - Vidyadip

Question

$\mathop {\lim }\limits_{n \to \infty } \frac{{n{{(2n + 1)}^2}}}{{(n + 2)({n^2} + 3n - 1)}} = $

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Answer

c
(c) $\mathop {\lim }\limits_{n \to \infty } \,\frac{{n\,{{(2n + 1)}^2}}}{{(n + 2)\,\,({n^2} + 3n - 1)}} = \mathop {\lim }\limits_{n \to \infty } \,\,\frac{{4{n^3} + 4{n^2} + n}}{{{n^3} + 5{n^2} + 5n - 2}}$

$ = \mathop {\lim }\limits_{n \to \infty } \frac{{{n^3}\,\left( {4 + \frac{4}{n} + \frac{1}{{{n^2}}}} \right)}}{{{n^3}\left( {1 + \frac{5}{n} + \frac{5}{{{n^2}}} - \frac{2}{{{n^3}}}} \right)}} = 4$

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