1.

If `f(x)={(x^3+x^2-16x+20)/(x-2)^2,x ne 0` k, x=0 is continuous for all real values of x, find the value of K.

Answer» x=0
f(0)=k
R.H.L `=underset(xrarr0^+)lim f(x)==underset(xrarr0)limf(0+h)`
`=underset(hrarr0)lim(h^3+h^2-16h+20)/(h-2)^2`
`=underset(hrarr0)lim((h-2)^2(h+5))/(h-2)^2`
`=underset(hrarr0)lim((h-5))/(1)=0+5=5`
`L.H.L =underset(xrarr0^-)limf(x)=underset(xrarr0)limf(0-h)`
`=underset(hrarr0)lim((-h)^2+(-h)^2-16(-h)+20)/(-h-2)^2`
`=underset(hrarr0)lim(-h^3+h^3+16h+20)/(h+2)^2`
`=underset(hrarr0)lim((h+2)^2(-h+5))/((h+5)^2)`
`=underset(hrarr0)lim((-h+5))/(1)=5`
`therefore`(fx) is continuous at x=0
`therefore` R.H.L=f(0)=L.H.L
`rArr` K=5.


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