Question

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

Answer 1

$limlimits_{ntoinfty}n(sqrt{n^2+2}-sqrt{n^2-2})=limlimits_{ntoinfty}frac{n(sqrt{n^2+2}-sqrt{n^2-2})(sqrt{n^2+2}+sqrt{n^2-2})}{sqrt{n^2+2}+sqrt{n^2-2}}\\=limlimits_{ntoinfty}frac{nleft[(sqrt{n^2+2})^2-(sqrt{n^2-2})^2]}{sqrt{n^2(1+frac{2}{n^2})}+sqrt{n^2(1-frac{2}{n^2})}}=limlimits_{ntoinfty}frac{n(n^2+2-n^2+2)}{nsqrt{1+frac{2}{n^2}}+nsqrt{1-frac{2}{n^2}}}$

$=limlimits_{ntoinfty}frac{ncdot4}{nleft(sqrt{1+frac{2}{n^2}}+sqrt{1-frac{2}{n^2}}right)}=limlimits_{ntoinfty}frac{4}{sqrt{1+frac{2}{n^2}}+sqrt{1-frac{2}{n^2}}}=frac{4}{sqrt1+sqrt1}\\=frac{4}{1+1}=frac{4}{2}=2$