First slide

Introduction to limits

Question

$The value of θ, is \lim_{θ \to 0} \frac{\cos^{2} \{1 - \cos^{2} (1 - \cos^{2} \dots . (1 - \cos^{2} \{1 - \cos^{2} θ\}))\}}{\sin (\frac{π \sqrt{θ + 4} - 2}{θ})}$

Moderate

A

$\frac{\sqrt{2}}{4}$
B

$\sqrt{2}$
C

1
D

2

Solution

$\begin{array}{l} \lim_{θ \to 0} \frac{\cos^{2} \{1 - \cos^{2} (1 - \cos^{2} \dots . (1 - \cos^{2} \{1 - \cos^{2} θ\}))\}}{\sin (\frac{π \sqrt{θ + 4} - 2}{θ})} \\ \lim_{θ \to 0} \frac{\cos^{2} \{1 - \cos^{2} (1 - \cos^{2} \dots . (1 - \cos^{2} \{1 - \cos^{2} θ\}))\}}{\sin (\frac{π (\sqrt{θ + 4} - 2)}{θ} . \frac{(\sqrt{θ + 4} + 2)}{(\sqrt{θ + 4} + 2)})} \\ \lim_{θ \to 0} \frac{1}{\sin [\frac{π}{\sqrt{θ + 4} + 2}]} = \frac{1}{\sin \frac{π}{4}} = \sqrt{2} \end{array}$

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