Question

If $f\left(x\right)=\frac{2x-{\sin }^{-1}x}{2x+{\tan }^{-1}x}$ is continuous at every point in its domain, then the value of $f\left(0\right)$ is

• A. $2$
• B. $-\frac{1}{3}$
• C. $\frac{2}{3}$
• D. $\frac{1}{3}$

Answer 1

The correct option is D $\frac{1}{3}$

For $f\left(x\right)$ to be continuous at every point of its domain, it must be continuous at $x=0.$
$\therefore$ We must have
$f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)$
$\Rightarrow f\left(0\right)=\underset{x\to 0}{lim}\frac{2x-{\sin }^{-1}x}{2x+{\tan }^{-1}x}$
Dividing numerator and denominator by $x$
$=\underset{x\to 0}{lim}\left\{\frac{2-\frac{{\sin }^{-1}x}{x}}{2+\frac{{\tan }^{-1}x}{x}}\right\}=\frac{2-1}{2+1}=\frac{1}{3}$
$\therefore f\left(0\right)=\frac{1}{3}$