830/VB01. zadatak / Matematika - Zbirka rešenih zadataka

MR-830 / 830. zadatak

$\underset{n \to \infty}{l i m} {(1 + \frac{2}{n + 1})}^{n}$

$\underset{n \to \infty}{l i m} {(1 + \frac{2}{n + 1})}^{n}$

$\frac{1}{t} = \frac{2}{n + 1}$

$t = \frac{n + 1}{2}$

$2 \cdot t = n + 1$

$n + 1 = 2 \cdot t$

$n = 2 \cdot t - 1$

$\begin{matrix} n \to \infty \\ t \to \infty \end{matrix}$

$\underset{n \to \infty}{l i m} {(1 + \frac{2}{n + 1})}^{n}$

$= \underset{t \to \infty}{l i m} {(1 + \frac{1}{t})}^{2 t - 1}$

$= \underset{t \to \infty}{l i m} [{(1 + \frac{1}{t})}^{2 t} \cdot {(1 + \frac{1}{t})}^{- 1}]$

$= \underset{t \to \infty}{l i m} [{(1 + \frac{1}{t})}^{t \cdot 2} \cdot {(1 + \frac{1}{t})}^{- 1}]$

$= {(1 + \frac{1}{\infty})}^{\infty \cdot 2} \cdot {(1 + \frac{1}{\infty})}^{- 1}$

$= e^{2} \cdot {(1 + 0)}^{- 1}$

$= e^{2}$

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