A) \[f(x)\]is continuous at \[x=0\]
B) \[f(x)\]is continuous at \[x=\pi \]
C) \[f(x)\]is continuous at \[x=\frac{3\pi }{4}\]
D) \[f(x)\]is discontinuous at \[x=\frac{3\pi }{4}\]
Correct Answer: C
Solution :
Here \[f\,\left( \frac{3\pi }{4} \right)=1\] and \[\because f\] \[\underset{x\to 3\pi /4+}{\mathop{\lim }}\,f(x)=\underset{h\to 0}{\mathop{\lim }}\,\,\,2\sin \frac{2}{9}\,\left( \frac{3\pi }{4}+h \right)=2\,\sin \frac{\pi }{6}=1\]. Hence \[f(x)\] is continuous at \[\frac{\sin \,\,2x}{\sin \,\left( \frac{x}{2} \right)}\].
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