Question

$\displaystyle\lim _{x \rightarrow 0} x \,\log \,\sin\, x$ is equal to :

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

Answer 1

Answer: (A)

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