If f(x) = l , , , , , , , , , , , , , ,1, , when , , , ,0 < x 3 4… - Vidyadip

MCQ

If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,1,\,{\rm{when\,\,}}\,\,0 < x \le \frac{{3\pi }}{4}\\2\sin \frac{2}{9}x,{\rm{when\,\,}}\,\frac{{3\pi }}{4} < x < \pi \end{array} \right.$, then

A
$f(x)$ is continuous at $x = 0$
B
$f(x)$ is continuous at $x = \pi $
✓
$f(x)$ is continuous at $x = \frac{{3\pi }}{4}$
D
$f(x)$ is discontinuous at $x = \frac{{3\pi }}{4}$

✓

Answer

Correct option: C.

$f(x)$ is continuous at $x = \frac{{3\pi }}{4}$

c
(c) Here $f\,\left( {\frac{{3\pi }}{4}} \right) = 1$ and

$\mathop {\lim }\limits_{x \to 3\pi /4 + } f(x) = \mathop {\lim }\limits_{h \to 0} \,\,2\sin \frac{2}{9}\,\left( {\frac{{3\pi }}{4} + h} \right) = 2\,\sin \frac{\pi }{6} = 1$.

Hence $f(x)$ is continuous at $x = \frac{{3\pi }}{4}$

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