Question

If $f\left(x\right)=\frac{x\sin 5x}{\tan 2x\tan 7x},x\ne 0,f\left(0\right)=\frac{5}{9}$ then at x = 0 f(x) is

• A. continuous
• B. discontinuous
• C. not determined
• D. none

Answer 1

The correct option is A continuous

$F\left(x\right)=\frac{x\sin 5x}{\tan 2x\tan 7x}$

$f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)\Rightarrow \underset{x\to 0}{lim}\frac{x\sin 5x}{\tan 2xtan7x}$

$\Rightarrow \underset{x\to 0}{lim}\frac{\frac{\sin 5x}{\left(5x\right)}x\left(5x\right)\left(x\right)}{\frac{\tan 2x}{2x}\times \left(2x\right)\frac{\tan \left(7x\right)}{7x}\left(7x\right)}$

$\Rightarrow \underset{x\to 0}{lim}\frac{5x^2}{14x^2}\times \frac{\sin 5x}{\left(5x\right)}\times \frac{1}{\left(\frac{\tan 2x}{2x}\right)\left(\frac{\tan 7x}{7x}\right)}$

$\Rightarrow \frac{5}{14}\times \left(1\right)\times \frac{1}{\left(1\right)\times \left(1\right)}\Rightarrow \frac{5}{14}$

$\therefore f\left(0\right)=\frac{5}{14}$

but given that $f\left(0\right)=\frac{5}{9}$

$\therefore \left(x\right)$ is discontinuous.