let u_n = (e −(1+(1/n))^n )^((√(n^2 +2)) −(√(n^2 +1))) find lim u


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Question Number 32364 by prof Abdo imad last updated on 23/Mar/18

$l e t u_{n} = {(e - {(1 + \frac{1}{n})}^{n})}^{\sqrt{n^{2} + 2} - \sqrt{n^{2} + 1}}$ $f i n d l i m u_{n}$
Commented by prof Abdo imad last updated on 06/Apr/18

$w e h a v e A_{n} = \sqrt{n^{2} + 2} - \sqrt{n^{2} + 1}$ $= n (\sqrt{1 + \frac{2}{n}} - \sqrt{1 + \frac{1}{n}}) b u t$ $\sqrt{1 + \frac{2}{n}} \sim 1 + \frac{1}{n} a n d \sqrt{1 + \frac{1}{n}} \sim 1 + \frac{1}{2 n} \Rightarrow$ $A_{n} \sim n + 1 - n - \frac{1}{2} \Rightarrow A_{n} \sim \frac{1}{2} a l s o w e h a v e$ ${(1 + \frac{1}{n})}^{n} = e^{n l n (1 + \frac{1}{n})}$ ${l n (1 + x))}^{^{'}} = \frac{1}{1 + x} = 1 - x + o (x^{2}) f o r x \in V (o)$ $l n (1 + x) = x - \frac{x^{2}}{2} + o (x^{3}) \Rightarrow$ $l n (1 + \frac{1}{n}) = \frac{1}{n} - \frac{1}{2 n^{2}} + o (\frac{1}{n^{3}}) \Rightarrow$ $n l n (1 + \frac{1}{n}) = 1 - \frac{1}{2 n} + o (\frac{1}{n^{2}}) \Rightarrow$ $e^{n l n (1 + \frac{1}{n})} = e^{1 - \frac{1}{2 n} + o (\frac{1}{n^{2}})} = e . (1 - \frac{1}{2 n} + o (\frac{1}{n^{2}})) \Rightarrow$ $e - {(1 + \frac{1}{n})}^{n} \sim \frac{e}{2 n} + o (\frac{1}{n^{2}}) \Rightarrow u_{n} \sim \sqrt{\frac{e}{2 n}} \Rightarrow$ ${l i m}_{n \to \infty} u_{n} = 0 .$
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