Question

$\underset{x\to 0}{\lim }x\log \sin x$ is equal to :

• A. zero
• B. $\infty$
• C. 1
• D. cannot be determined

Answer 1

Answer: (A)

Given, $\underset{x\to 0}{\lim }x\log \sin x$
$=\underset{x\to 0}{\lim }\frac{\log \sin x}{1/x}(\frac{\infty }{\infty }$ form)
By using L-Hospital's rule, we get
$=\underset{x\to 0}{\lim }\frac{(1/\sin x)\cos x}{-1/x^2}=\underset{x\to 0}{\lim }-\frac{\cos x\cdot x}{\frac{\sin x}{x}}$
$=\frac{0}{1}=0$
$\therefore$ Required limits is 0