Question

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

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

Answer 1

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

For $f\left(x\right)$ to be continuous at every point of its domain, it must be continuous at $x=0$ also,
$\therefore$ 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 $x$
$=\underset{x\to 0}{lim}\left\{\frac{2-\frac{{\sin }^{-1}x}{x}}{2+\frac{{\tan }^{-1}x}{x}}\right\}=\frac{1}{3}$

$\therefore f\left(0\right)=\frac{1}{3}$