Question

The function $f\left(x\right)=\frac{x\tan 2x}{\sin 3x.\sin 5x}$ for $x\ne 0$, $f\left(0\right)=2/17$ at $x=0$ is:

• A. continuous
• B. discontinuous
• C. differentiable
• D. not differentiable

Answer 1

The correct option is B discontinuous

$f\left(x\right)=\frac{x\tan 2x}{\sin 3x.\sin 5x}$
${lim}_{x\to 0}f\left(x\right)={lim}_{x\to 0}\frac{x\frac{\tan 2x}{2x}.2x}{15x^2\frac{\sin 3x}{3x}.\frac{\sin 5x}{5x}}$
$={lim}_{x\to 0}\frac{2x^2}{15x^2}=\frac{2}{15}$
Given $f\left(0\right)=\frac{2}{17}$
Here , ${lim}_{x\to 0}f\left(x\right)\ne f\left(0\right)$
Hence, $f\left(x\right)$ is discontinuous at $x=0$