Question

Let `f(x)={x}=` greater integer less than or eqal to x. For any integer k, show that `lim_(xtok) f(x)` does not exist.

Answer 1

We have
`underset(xtok^(+))limf(x)=underset(hto0)limf(k+h)=underset(hto0)lim[k+h]=underset(hto0)limk=k{because[k+h]=k}`
` and underset(xtok^(-))limf(x)=underset(hto0)lim(k-h)=underset(hto0)lim[k-h]=underset(hto0)lim(k-1)=(k-1)[because[k-h]=k-1]`
Thus, `underset(xtok)limf(x)neunderset(xtok^(-))lim f(x).`
Hence,`underset(xtok)limf(x)` does not exist.