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

Question Number 63267 by Tawa1 last updated on 01/Jul/19

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

Commented by Tawa1 last updated on 01/Jul/19

God bless you sir

Commented by mathmax by abdo last updated on 01/Jul/19

l e t A_{n} = {(\frac{n^{3} + 1}{n^{3} - 1})}^{2 n - n^{3}} \Rightarrow A_{n} = {(1 - \frac{2}{n^{3} - 1})}^{2 n - n^{3}} = e^{(2 n - n^{3}) l n (1 - \frac{2}{n^{3} - 1})} w e h a v e

l n (1 - x) = - x + o (x^{2}) \Rightarrow l n (1 - x) \sim - x (x \to 0) \Rightarrow l n (1 - \frac{2}{n^{3} - 1}) \sim - \frac{2}{n^{3} - 1} (n \to + \infty)

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

{l i m}_{n \to + \infty} A_{n} = e^{2} .

Commented by mathmax by abdo last updated on 02/Jul/19

y o u a r e w e l c o m e .