1.

Which of the following function are continuous at x=0 ? [Note : sgn x denotes signum dunction od x.]

Answer»

`COS((pi)/(2)sgn|X|)+sgn|x|`
`cos((pi)/(2)sgn|x|)-sgn|x|`
`SIN((pi)/(2)sgn|x|)+sgn|x|`
`sin((pi)/(2)sgn|x|)-sgn|x|`

Solution :We have `cos((pi)/(2)sgn|x|)={{:(0","x!=0),(1","x=0):}`
`"As,"sgn|x|={{:(0","x!=0),(1","x=0):}rArrcos((pi)/(2)sgn|x|)+sgn|x|=1AAx INR`
`"Also,"cos((pi)/(2)sgn|x|)-sgn|x|={{:(-1","x!=0),(1","x=0):}`
`"As,"sin((pi)/(2)sgn|x|)-sgn|x|={{:(1","x!=0),(0","x=0):}rArrsin((pi)/(2)sgn|x|)+sgn|x|={{:(2","x!=0),(0","x=0):}`
`andsin((pi)/(2)sgn|x|)-sgn|x|=0AAx inR`.


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