Question

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

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

Answer 1

Answer: (B)

$\frac{1}{3}$

The function f is clearly continuous at each point in its domain except possibly at $x=0$. Given that f(x) is continuous at $x=0$.
$\therefore f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)$
$=\underset{x\to 0}{lim}\frac{2x-si{n}^{-1}x}{2x+ta{n}^{-1}x}$
$=\underset{x\to 0}{lim}\frac{2-\left(si{n}^{-1}x\right)/x}{2+\left(ta{n}^{-1}x\right)x}$
$=\frac{1}{3}$