Question

Помогите решить предел)

Answer 1

$limlimits_{n to infty} ( sqrt{(n^2+1)(n^2+2)}- sqrt{(n^2-1)(n^2-2)} )=\\= limlimits _{n to infty}frac{(n^2+1)(n^2+2)-(n^2-1)(n^2-2)}{ sqrt{(n^2+1)(n^2+2)}+sqrt{(n^2-1)(n^2-2)}}=\\= limlimits _{n to infty} frac{n^4+3n^2+2-(n^4-3n^2+2)}{sqrt{n^4+3n^2+2}+sqrt{n^2-3n^2+2}} =Big [frac{:n^2}{:n^2}Big ]=\\= limlimits _{n to infty}frac{frac{6n^2}{n^2}}{sqrt{1+frac{3}{n^2}+frac{2}{n^4}}+sqrt{1-frac{3}{n^2}+frac{2}{n^4}}}=frac{6}{sqrt1+sqrt1}=frac{6}{2}=3$