First slide

Introduction to limits

Question

The value of $lim_{x \to 0} \frac{(1 - \cos 2 x) \sin 5 x}{x^{2} \sin 3 x}$ is

Moderate

A

$\frac{10}{3}$
B

$\frac{3}{10}$
C

$\frac{6}{5}$
D

$\frac{5}{6}$

Solution

$\begin{matrix} lim_{x \to 0} \frac{(1 - \cos 2 x) \sin 5 x}{x^{2} \sin 3 x} = lim_{x \to 0} \frac{2 \sin^{2} xsin 5 x}{x^{2} \sin 3 x} \\ = lim_{x \to 0} (\frac{2 \sin^{2} x}{x^{2}}) \frac{(\frac{\sin 5 x}{x})}{(\frac{\sin 3 x}{x})} \\ = lim_{x \to 0} 2 {(\frac{\sin x}{x})}^{2} \times \frac{5 lim_{x \to 0} (\frac{\sin 5 x}{5 x})}{3 lim_{x \to 0} (\frac{\sin 3 x}{3 x})} \\ = \frac{2 \times 5}{3} = \frac{10}{3} [∵ lim_{x \to 0} \frac{\sin x}{x} = 1] \end{matrix}$

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