Limits Continuity And Differentiability Question 117
Question: If $ f(x)=\cos [ \frac{\pi }{x} ]\cos ( \frac{\pi }{2}(x-1) ); $ where [x] is the greatest integer function of x, then f(x) is continuous at
Options:
A) x = 0
B) x = 1, 2
C) x = 0, 2, 4
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Clearly f(x) is not continuous at x = 0 Now $ f(1)=cos3 $
$ \underset{x\to {1^{-}}}{\mathop{\lim }}f(x)=\underset{h\to 0}{\mathop{\lim }}\cos 3\cos { \frac{\pi }{2}(-h) }=\cos 3 $
$ \underset{x\to {1^{+}}}{\mathop{\lim }}f(x)=\underset{h\to 0}{\mathop{\lim }}\cos 3\cos { \frac{\pi }{2}(h) }=\cos 3 $ Further $ f(2)=cos1.cos\frac{\pi }{2}=0 $
$ \underset{x\to {2^{-}}}{\mathop{\lim }}f(x)=\underset{h\to 0}{\mathop{lim}}cos1.cos{ \frac{\pi }{2}(1-h) } $
$ =\underset{h\to 0}{\mathop{\lim }}\cos 1.0=0 $
$ =\underset{x\to {2^{+}}}{\mathop{\lim }}f(x)=\underset{h\to 0}{\mathop{\lim }}\cos 1.\cos { \frac{\pi }{2}(1+h) }=0 $
$ \therefore f(x) $ is continuous at $ x=1 $ and $ x=2 $ both
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