If `("lim")_(xto1^-)(2-x+a[x-1]+b[1+x])` exists, then `aa n db` can take the values of (where [.] denotes the greatest integer function). (a)`a=1/3,b=1` (b) `a=1, b=-1` (c) `a=9, b=-9` (d)`a=2, b=2/3`

Answer

B, C

Explanation

Since the greatest integer function is discountinuous at integral values of x, for a given limit to exist both left-and right-hand limits must be equal.
`L.H.L.=underset(xto1^(-))lim(2-x+a[x-1]+b[1+x])`
`=2-1+a(-1)+b(1)=1-a+b`
`R.H.L.=underset(xto1^(+))lim(2-x+a[x-1]+b[1+x])`
`=2-1+a(0)+b(2)=1+2b`
On comparing, we have `-a=b`.