First slide

Introduction to limits

Question

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

Moderate

A

$\frac{1}{2}$
B

$\frac{1}{4}$
C

$\frac{1}{8}$
D

none of these

Solution

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

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