Giải bài tập 5.2 trang 78 SBT toán 11 tập 1 kết nối

a) $\lim_{n \to + \infty}(\sqrt{n^{2}+2n}-n-2)$

$=\lim_{n \to +\infty}\frac{-2n-4}{\sqrt{n^{2}+2n}+n+2}$

$=\lim_{n \to +\infty}\frac{-2-\frac{4}{n}}{\sqrt{1+\frac{2}{n}}+1+\frac{2}{n}}=-1$

b) $\lim_{n \to + \infty}(2+n^{2}-\sqrt{n^{4}+1})$

$=\lim_{n \to +\infty}\frac{4n^{2}+3}{2+n^{2}+\sqrt{n^{4}+1}}$

$=\lim_{n \to +\infty}\frac{4+\frac{3}{n^{2}}}{\frac{2}{n^{2}}+1+\sqrt{1+\frac{1}{n^{4}}}}$

$=2$

c) $\lim_{n \to + \infty}(\sqrt{n^{2}-n+2}+n)$

$=\lim_{n \to +\infty}n(\sqrt{1-\frac{1}{n}+\frac{2}{n^{2}}}+1)$

$=+\infty$

d) $\lim_{n \to +\infty}(3n - \sqrt{4n^{2}+1})$

$=\lim_{n \to +\infty}n(3-\sqrt{4+\frac{1}{n^{2}}})$

$=+\infty$