First slide

Introduction to limits

Question

The value of $lim_{n \to \infty} {(\cos \frac{x}{n})}^{n}$ , is

Moderate

A

$e$
B

$e^{- 1}$
C

$1$
D

none of these

Solution

we have,
$\begin{array}{l} lim_{n \to \infty} {(\cos \frac{x}{n})}^{n} \\ = lim_{n \to \infty} {\{1 + (\cos \frac{x}{n} - 1)\}}^{n} \\ = lim_{n \to \infty} {\{1 - 2 \sin^{2} \frac{x}{2 n}\}}^{n} \end{array}$
$= e^{lim_{n \to \infty} - 2 n \sin^{2} \frac{x}{2 n}}$
$= e^{- lim_{n \to \infty} {(\frac{\sin \frac{x}{2 n}}{x / 2 n})}^{2} \times \frac{1}{2 n}}$
$= e^{- 1 \times 0} = e^{0} = 1$

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