First slide

Introduction to limits

Question

$lim_{x \to 0} \frac{(1 - \cos 2 x) \sin 5 x}{x^{2} \sin 3 x}$ equals

Moderate

A

$10 / 3$
B

$3 / 10$
C

$6 / 5$
D

$5 / 6$

Solution

We have,
$\begin{array}{l} lim_{x \to 0} \frac{(1 - \cos 2 x) \sin 5 x}{x^{2} \sin 3 x} \\ = lim_{x \to 0} \frac{1 - \cos 2 x}{x^{2}} \cdot lim_{x \to 0} \frac{\sin 5 x}{\sin 3 x} \\ = lim_{x \to 0} 2 {(\frac{\sin x}{x})}^{2} \cdot lim_{x \to 0} (\frac{\sin 5 x}{5 x} \cdot \frac{3 x}{\sin 3 x} \cdot \frac{5 x}{3 x}) = 2 \times \frac{5}{3} = \frac{10}{3} \end{array}$

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