lim-n-n-1-3-n-2-1-1-3-2n-2-3n-1-1-2-

Question Number 122866 by bemath last updated on 20/Nov/20

\lim_{n \to \infty} n^{\frac{1}{3}} \sqrt[3]{n^{2} + 1} {(2 n^{2} + 3 n - 1)}^{- \frac{1}{2}} .

Answered by liberty last updated on 20/Nov/20

\lim_{x \to \infty} \frac{\sqrt[3]{x^{3} + x}}{\sqrt{2 x^{2} + 3 x - 1}} = \lim_{x \to \infty} \frac{\sqrt[3]{1 + x^{- 2}}}{\sqrt{2 + 3 x^{- 1} - x^{- 2}}}

= \frac{1}{\sqrt{2}} .

Answered by mathmax by abdo last updated on 21/Nov/20

u_{n} = n^{\frac{1}{3}} {(n^{2} + 1)}^{\frac{1}{3}} {({2 n}^{2} + 3 n - 1)}^{- \frac{1}{2}} \Rightarrow

u_{n} = n^{\frac{1}{3}} n^{\frac{2}{3}} {(1 + \frac{1}{n^{2}})}^{\frac{1}{3}} 2^{- \frac{1}{2}} . n^{- 1} {(1 + \frac{3}{2 n} - \frac{1}{{2 n}^{2}})}^{- \frac{1}{2}}

= 2^{- \frac{1}{2}} {(1 + \frac{1}{n^{2}})}^{\frac{1}{3}} {(1 + \frac{3}{2 n} - \frac{1}{{2 n}^{2}})}^{- \frac{1}{2}}

\Rightarrow u_{n} \sim \frac{1}{\sqrt{2}} (1 + \frac{1}{{3 n}^{2}}) (1 - \frac{1}{2} (\frac{3}{2 n} - \frac{1}{{2 n}^{2}})) \Rightarrow

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