Evaluate the left-and right-hand limits of the function defined by `f(x)={1+x^2,if0lt=x<1 2-x ,ifx >1` at `x=1.` Also, show that `("lim")_(xvec1)f(x)` does not exist

Explanation

LHL of `f(x)` at x=1 is
`underset(xto1^(-))limf(x)=underset(hto0)f(1-h)`
`=underset(hto0)lim[1+(1-h)^(2)]`
`=underset(hto0)lim(2-2h+h^(2))=2`
RHL of f(x) at x=1 is
`underset(xto1^(+))limf(x)=underset(hto0)limf(1+h)`
`=underset(hto0)lim[2-(1+h)]`
`underset(hto0)lim(1-h)=1`
Clearly, `underset(xto1^(-))limf(x)neunderset(xto1^(+))f(x)`
So, `underset(xto1)limf(x)` does not exist.