Question

a)lim \(\frac{\left(2n+1\right)^2\left(n-1\right)}{\sqrt[3]{n^3+7n-2}}\)

b)lim [(2n-1)\(\sqrt{\frac{2n^2+5}{n^4+n^2+2}}\)]

c)lim [n(\(\sqrt[3]{n^3+n^2}-n\))]

Answer 1

a) lim \(\frac{\left(2n+1\right)^2\left(n-1\right)}{\sqrt[3]{n^3+7n-2}}\)

= lim \(\left(2n+1\right)^2.\frac{\left(1-\frac{1}{n}\right)}{\sqrt[3]{1+\frac{7}{n^2}-\frac{2}{n^3}}}\)

\(=+\infty\)

b) lim \(\left(2n-1\right)\sqrt{\frac{2n^2+5}{n^4+n^2+2}}\)

= lim \(\left(2-\frac{1}{n}\right)\sqrt{\frac{2+\frac{5}{n^2}}{1+\frac{1}{n^2}+\frac{2}{n^4}}}\)

=2.2 = 4

c ) = lim \(n.\frac{n^2}{\sqrt[3]{\left(n^3+n^2\right)^2+n\sqrt[3]{n^3+n^2}+n^2}}\)

= lim \(n.\frac{1}{\sqrt[3]{\left(1+\frac{1}{n}\right)^2+\sqrt[3]{1+\frac{1}{n}}+1}}\)

\(=+\infty\)