Evaluate the left- and right-hand limits of the function defined `f(x)={{:(1+x^(2)",if "0lexlt1),(2-x", if" xgt1):}"at "x=1. "Also, show that `lim_(xto1) 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.