A) \[\frac{\pi }{5}\]
B) \[\frac{5}{\pi }\]
C) 1
D) 0
Correct Answer: A
Solution :
Since \[f(x)\] is continuous at \[x=0,\] therefore \[\underset{x\to 0}{\mathop{\text{lim}}}\,f(x)=f(0)\] Þ \[\underset{x\to 0}{\mathop{\text{lim}}}\,\frac{\sin \pi \,x}{5x}=k\] Þ \[\underset{x\to 0}{\mathop{\text{lim}}}\,\left( \frac{\sin \pi \,x}{\pi x} \right)\,.\,\frac{\pi }{5}=k\] Þ \[(1)\,.\,\frac{\pi }{5}=k\] Þ \[k=\frac{\pi }{5}\].
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