Tính $\lim \frac{{n\sqrt {1 + 3 + 5 + ... + \left( {2n - 1} \right)} }}{{2{n^2} + 1}}$.

$\lim \frac{{n\sqrt {1 + 3 + 5 + ... + \left( {2n - 1} \right)} }}{{2{n^2} + 1}}$

Xét 1 + 3 + 5 +… + (2n – 1) có số số hạng là: (2n – 1 – 1) : 2 + 1 = n

$\lim \frac{{n\sqrt {1 + 3 + 5 + ... + \left( {2n - 1} \right)} }}{{2{n^2} + 1}}$

$= \lim \frac{{n\sqrt {\frac{{2n.n}}{2}} }}{{2{n^2} + 1}}$

$= \lim \frac{{n\sqrt {{n^2}} }}{{2{n^2} + 1}}$

$= \lim \frac{{{n^2}}}{{2{n^2} + 1}}$

$= \lim \frac{1}{{2 + \frac{1}{{{n^2}}}}}$

$= \frac{1}{2}$.