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lim(xto1) (xsin(x-[x]))/(x-1), where [.]...

`lim_(xto1) (xsin(x-[x]))/(x-1)`, where `[.]` denotes the greatest integer function, is equal to

A

`(2)/(pi-1)`

B

`(pi+1)/(2)`

C

`(2)/(pi+1)`

D

`(2(pi+1))/(pi-1)`

Text Solution

Verified by Experts

The correct Answer is:
C
`underset(xto1)lim(xsin(x-[x]))/(x-1)`
Now,`L.H.L.=underset(hto0)lim((1-h)sin(1-h-[1-h]))/((1-h)-1)`
`=underset(hto0)lim((1-h)sin(1-h))/(-h)=-oo`
`R.H.L.=underset(hto0)lim((1+h)sin(1+h-[1+h]))/((1+h)-1)=underset(hto0)lim((1+h)sinh)/(h)=1`
Hence, the limit does not exist.
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