If f(x) = l , , , , , , , , , , , ,x x, , when , , 0 < x 2 2 ( + x), when , , 2 < x < ., then

If f(x) = l , , , , , , , , , , , ,x x, , when , , 0 < x 2 2 ( + …

MCQ

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

A
$f(x)$ is discontinuous at $x = \pi /2$
B
$f(x)$ is continuous at $x = \pi /2$
C
$f(x)$ is continuous at $x = 0$
D
None of these

✓

Answer

$\mathop {\lim }\limits_{x \to \frac{\pi }{{{2^ - }}}} \,f(x) = \frac{\pi }{2}, \mathop {\lim }\limits_{x \to \frac{\pi }{{{2^ + }}}} f(x) = \frac{{ - \pi }}{2}$
and $f\left( {\frac{\pi }{2}} \right) = \frac{\pi }{2}.$

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