First slide

Introduction to limits

Question

The value of $lim_{x \to 0} \frac{\sin x - x + \frac{x^{3}}{6}}{x^{5}}$ is

Moderate

A

0
B

1
C

$1 / 60$
D

$1 / 120$

Solution

We have,
$\begin{array}{l} lim_{x \to 0} \frac{\sin x - x + \frac{x^{3}}{6}}{x^{5}} \\ = lim_{x \to 0} \frac{x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots - x + \frac{x^{3}}{6}}{x^{5}} \end{array}$
$= lim_{x \to 0} \frac{1}{5!} - \frac{x^{2}}{7!} + \dots = \frac{1}{120}$

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