First slide

Introduction to limits

Question

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

Moderate

A

$1 / 2$
B

$- 1 / 2$
C

0
D

1

Solution

We have,
$\begin{array}{l} lim_{x \to 0} \frac{1 + \sin x - \cos x + \log (1 - x)}{x^{3}} \\ = lim_{x \to 0} \frac{1 + (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots) - (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} \dots) + (- x - \frac{x^{2}}{2} - \frac{x^{3}}{3} - \frac{x^{4}}{4})}{x^{3}}) \\ = lim_{x \to 0} (- \frac{1}{3!} - \frac{1}{3}) + x^{2} (\frac{1}{5!} - \frac{1}{5}) + \dots = - \frac{1}{6} - \frac{1}{3} = - \frac{1}{2} \end{array}$

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