If `f(x)= {{:(mx^(2)+n,xlt0),(nx+m,0ltxle1),( nx^(3)+m,xgt1):}`
For what integers m and n does both `lim_(xrarr0) f(x)` and `lim_(xrarr1) f(x)` exist?

Explanation

at x=0
`LHL=underset(xrarr0^(-))"lim"f(x)`
`=underset(hrarr0)"lim"f( 0-h)`
`=underset(hrarr0)"lim"m( 0-h)^(2)+n=n`
RHL`=underset(xrarr0^(+))"lim"f(x)`
`=underset(hrarr0)"lim"f(0+h)`
`=underset(hrarr0)"lim"n(0+h)+m=m`
`because underset(xrarr0)"lim"f(x)` exists.
Let `0- h=x`
`rArr0-hrarr0`
`rArr hrarr0`
Let `0+h=x`
`rArr 0+hrarr0`
`rArr hrarr0`