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If f(x) =[m x^2+n , x<0n x+m , xlt=xlt=1...

If `f(x) =[m x^2+n , x<0n x+m , xlt=xlt=1n x^3+m , x >1`. For what integers m and n does both `(lim)_(x->1)f(x)dot`

Text Solution

Verified by Experts

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`
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