First slide

Introduction to limits

Question

$lim_{m \to \infty} {(\cos \frac{x}{m})}^{m}$ is equal to

Moderate

A

0
B

e
C

$\frac{1}{e}$
D

1

Solution

$\begin{array}{l} lim_{m \to \infty} {(\cos \frac{x}{m})}^{m} = lim_{m \to \infty} {[1 + (\cos \frac{x}{m} - 1)]}^{m} \\ = lim_{m \to \infty} {[1 - (1 - \cos \frac{x}{m})]}^{m} \\ = lim_{m \to \infty} {[1 - 2 \sin^{2} \frac{x}{2 m}]}^{m} \\ = e^{lim_{m \to \infty} m [- 2 \sin^{2} \frac{x}{2 n}]} \\ \underset{m \to \infty}{lim - 2} {\{\frac{\sin \frac{x}{2 m}}{x^{2}})}^{2} (\frac{x^{2}}{4 m^{2}}) m \\ = e \\ = e^{- 2 lim_{m \to \infty} \frac{x^{2}}{4 m}} = e^{0} = 1 \end{array}$

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