First slide

Introduction to limits

Question

The value of $lim_{x \to 0} \frac{1 - \cos^{3} x}{x \sin x \cos x}$ , is

Moderate

A

$\frac{2}{5}$
B

$\frac{3}{5}$
C

$\frac{3}{2}$
D

$\frac{3}{4}$

Solution

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

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