Let `f(x)={{:(mx^(2)+n",",xlt0),(nx+m",",0lexle1),(nx^(3)+m",",xgt1):}`
For what vlaues of intergeras of integers a and n, `lim_(xto0) f(x)and lim_(xto1) f(x)` both exist?

Explanation

We have
`underset(xto0^(+))limf(x)=underset(hto0)limf(0+h)underset(hto0)limf(h)=underset(hto0)lim(nh+m)=m.`
`underset(xto0^(-))limf(x)=underset(hto0)limf(0-h)=underset(hto0)limf(-h)=underset(hto0)lim"{"m(-h)^(2)+n"}"=underset(hto0)lim(mh^(2)+n)=n.`
` therefore underset(xto0)limf(x)` exist only when `m=n.`
Now, `underset(xto1^(+))limf(x)=underset(hto0)limf(1+h)=underset(xto0)lim"{"n(12+h)^(3)+m"}"=underset(hto0)lim{n(1+h^(3)+3h+3h^(2))+m}=(n+m).`
And, `underset(xto1^(-))limf(x)=underset(hto0)limf(1-h)=underset(hto0)lim{n(1-h)+m}=(n+m).`
`thereforeunderset(xto1)limf(x)=(n+m).`
Hence `underset(xto0)f(x)and underset(xto1)limf(x)` both exist only when `m=n.`