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("lim")(xvec1)(xsin(x-[x]))/(x-1),w h e ...

`("lim")_(xvec1)(xsin(x-[x]))/(x-1),w h e r e[dot]` denotes the greatest integer function is equal to (a) 0 (b) `-1` (c) not exist (d) none of these

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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