Question

For the function $f\left(x\right)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}$. The value of $f\left(\pi \right)$, so that $f\left(x\right)$ is continuous at $x=\pi$ is

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

Answer 1

The correct option is A $-1$

For continuity of $f\left(x\right)$ at $x=\pi$, we must have
$f\left(\pi \right)=\underset{x\to \pi }{lim}f\left(x\right)$
$\underset{x\to \pi }{lim}f\left(x\right)=\underset{x\to \pi }{lim}\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\left[\frac{0}{0}\text{Form}\right]$
Applying L Hospital's rule
$=\underset{x\to \pi }{lim}\frac{-\cos x-\sin x}{\cos x-\sin x}$
$=\frac{-\cos \pi -\sin \pi }{\cos \pi -\sin \pi }$
$=\frac{1-0}{-1-0}=-1$
$\therefore f\left(\pi \right)=-1$