Question

IF `f(x)={{:(x,"if",x,"is rational"),(1-x,"if",x, "is irrational"):},"then find" lim_(xto1//2) f(x)` if exists.

Answer 1

`underset(xto1//2^(+))limf(x)=underset(xto1//2^(+))limx" "`(if `1//2^(+)`rational)
`=1//2`
`underset(xto1//2^(-))limf(x)=underset(xto1//2^(+))lim(1-x)" "`(if `1//2^(+)`is irrational)
`=1-1//2=1//2`
So, `underset(xto1//2^(+))limf(x)=1//2` in any case.
Similarly, we get `underset(xto1//2^(-))limf(x)=1//2`
`:." "underset(xto1//2)limf(x)=1//2`