If f (x)= l x x ; 0 < x 2 2 ( +x) ; 2 < x< ., then - Vidyadip

MCQ

If $f (x)=\left\{\begin{array}{l}x \sin x ; 0 < x \leq \frac{\pi}{2} \\ \frac{\pi}{2} \sin (\pi+x) ; \quad \frac{\pi}{2}< x< \pi\end{array}\right.$, then

✓
$f (x)$ is discontinuous at $x=\frac{\pi}{2}$
B
$f (x)$ is continuous at $x=\frac{\pi}{2}$
C
$f (x)$ is continuous at $x=0$
D
none of these

✓

Answer

Correct option: A.

$f (x)$ is discontinuous at $x=\frac{\pi}{2}$

(A)
$\lim _{x \rightarrow \frac{\pi^{-}}{2}} f (x)=\lim _{x \rightarrow \frac{\pi}{2}} x \sin x=\frac{\pi}{2}$
$\lim _{x \rightarrow \frac{\pi^{+}}{2}} f(x)=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\pi}{2} \sin (\pi+x)=\frac{-\pi}{2}$
$\therefore f (x)$ is discontinuous at $x=\frac{\pi}{2}$.

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