First slide

Introduction to limits

Question

$lim_{n \to \infty} {\{{(\frac{n}{n + 1})}^{α} + \sin \frac{1}{n}\}}^{n},$ where $α \in Q$ is equal to

Moderate

A

$e^{- α}$
B

$- α$
C

$e^{1 - α}$
D

$e^{1 + α}$

Solution

$lim_{n \to c} {\{{(\frac{n}{n + 1})}^{α} + \sin \frac{1}{n}\}}^{n}$
$= lim_{n \to \infty} {\{{(1 + \frac{1}{n})}^{- α} + \sin \frac{1}{n}\}}^{n}$
$\begin{array}{l} = lim_{n \to \infty} {\{\{1 - \frac{α}{n} + \frac{α (α + 1)}{2!} \cdot \frac{1}{n^{2}} \dots\} + \{\frac{1}{n} - \frac{1}{3! n^{3}} + \frac{1}{5! n^{5}} - \dots\}\}}^{n} \\ = lim_{n \to \infty} {\{1 + (\frac{1 - α}{n}) + \frac{α (α + 1)}{2! n^{2}} \dots\}}^{n} \end{array}$
$= e^{\lim_{a \to \infty} \{(\frac{1 - α}{n}) + \frac{α (α + 1)}{2! n^{2}} + . . . .\} n} = e^{1 - α}$

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