First slide

Introduction to limits

Question

The value of $lim_{x \to 0} \frac{\sin [\log_{e} (1 + x)]}{\log (1 + \sin x)}$ is equal to

Moderate

A

0
B

$\frac{1}{2}$
C

$\frac{1}{4}$
D

1

Solution

$lim_{x \to 0} \frac{\sin [\log_{e} (1 + x)]}{\log (1 + \sin x)}$
It is an indeterminate $(\frac{0}{0})$ form.
$\begin{array}{l} = lim_{x \to 0} \frac{\cos [\log_{e} (1 + x)] \cdot \frac{1}{(1 + x)} \cdot (1)}{\frac{1}{(1 + \sin x)} (0 + \cos x)} \\ = \frac{\cos (\log_{e} 1) \cdot (1)}{1 \cdot \cos 0} = \frac{\cos 0}{\cos 0} = \frac{1}{1} = 1 \end{array}$

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