Question

Lim(n-> бесконечность) n*(sqrt((n^2)+2)-sqrt((n^2)-2))

Answer 1

$\lim\limits_{n\to\infty}n(\sqrt{n^2+2}-\sqrt{n^2-2})=\lim\limits_{n\to\infty}\frac{n(\sqrt{n^2+2}-\sqrt{n^2-2})(\sqrt{n^2+2}+\sqrt{n^2-2})}{\sqrt{n^2+2}+\sqrt{n^2-2}}\\\\=\lim\limits_{n\to\infty}\frac{n\left[(\sqrt{n^2+2})^2-(\sqrt{n^2-2})^2]}{\sqrt{n^2(1+\frac{2}{n^2})}+\sqrt{n^2(1-\frac{2}{n^2})}}=\lim\limits_{n\to\infty}\frac{n(n^2+2-n^2+2)}{n\sqrt{1+\frac{2}{n^2}}+n\sqrt{1-\frac{2}{n^2}}}$

$=\lim\limits_{n\to\infty}\frac{n\cdot4}{n\left(\sqrt{1+\frac{2}{n^2}}+\sqrt{1-\frac{2}{n^2}}\right)}=\lim\limits_{n\to\infty}\frac{4}{\sqrt{1+\frac{2}{n^2}}+\sqrt{1-\frac{2}{n^2}}}=\frac{4}{\sqrt1+\sqrt1}\\\\=\frac{4}{1+1}=\frac{4}{2}=2$