Question

Let $f\left(x\right)=\{\begin{array}{cc}\frac{1-\tan x}{4x-\pi },& x\ne \frac{\pi }{4}\\ \lambda ,& x=\frac{\pi }{4}\end{array}$. and $xϵ[0,\frac{\pi }{2})$

If $f\left(x\right)$ is continuous in $[0,\frac{\pi }{2})$ then $\lambda$ is

• A. $1$
• B. $\frac{1}{2}$
• C. $-\frac{1}{2}$
• D. none of these

Answer 1

The correct option is C $-\frac{1}{2}$

$f\left(x\right)=\underset{x\to \frac{\pi }{4}}{lim}\frac{1-\tan \left(x\right)}{4x-\pi }$

Applyig L'Hospital Rule,

$f\left(x\right)=\frac{-{\sec }^{2}x}{4}=\frac{-2}{4}=-1/2$