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Prove that the function f defined by f(x) ={{:((x)/(|x|+2x^(2)), if x ne 0),(k,if x = 0):}remainsdiscontinuous at x= 0,regardingsthe choice iof k.

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Solution :We have, `f(x) ={{:((x)/(|x|+2x^(2)), if x ne 0),(K,if x = 0):}`
At `x= 0, LHL =underset(xrarr0^(-))(lim)(x)/(|x|+2x^(2)) = underset(hrarr0)(lim)((0-h))/(|0-h|+2(0-h)^(2))`
`= underset(hrarr0)(lim)(-h)/(h+2h^(2))=underset(hrarr0)(lim)(-h)/(h(1+2h)) = -1`
`RHL= underset(xrarr0^(+))lim(x)/(|x|+2x^(2))= underset(hrarr0)(lim)(0+h)/(|0+h|+2(0+h)^(2))`
`= underset(hrarr0)(lim)(h)/(h+2h)^(2)=underset(hrarr0)(lim)(h)/(h(1+2h)) = 1`
and`f(0) =k`
Since, ` LHL neRHL` for anyvalue of k.
Hence, `f(x)` is discountinuousat `x = 0`regardiess the choiceofk.


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