Question

Show that the function $F\left(x\right)=|\sin x+\cos x|$ is continuous at $x=\pi$.

Answer 1

Use the concept of continuity of a function at a point
We have,
$F\left(x\right)=|\sin x+\cos x|$
If $F\left(a\right)=\underset{x\to a}{lim}F\left(x\right)$,
then $F\left(x\right)$ is continuous at $x=a$ $\underset{x\to \pi }{lim}F\left(x\right)=\underset{x\to \pi }{lim}(\sin x+\cos x)$ $=|\sin \pi +\cos \pi |$ $=|0-1|=1$ Also, $F\left(\pi \right)=|\sin \pi +\cos \pi |=|0-1|=1$ $\therefore F\left(\pi \right)=\underset{x\to \pi }{lim}f\left(x\right)$
Hence $F\left(x\right)$ is continuous at $x=\pi$