First slide

Introduction to limits

Question

$lim_{x \to 0} x \log_{e} (\sin x)$ is equal to

Moderate

A

$- 1$
B

$\log_{e} 1$
C

1
D

none of these

Solution

We have,
$lim_{x \to 0} x \log_{e} \sin x$ $[0 \times \infty form]$
$= lim_{x \to 0} \frac{\log_{e} \sin x}{1 / x} [\frac{\infty}{\infty} form]$
$= lim_{x \to 0} \frac{\cot x}{- 1 / x^{2}}$ [Using L' Hospital's rule]
$= - lim_{x \to 0} \frac{x^{2}}{\tan x} = - lim_{x \to 0} \frac{x}{\tan x} \times x = - (1 \times 0) = 0 = \log_{e} 1$

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