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Discuss the continuity of the following ...

Discuss the continuity of the following functions at the points shown against them :
`{:(f(x)=(x)/(|x|)",","for"x ne0),(=1",","for"x=0):}}at x=0.`

Text Solution

Verified by Experts

`f(0)=1" "...."Given"...(1)`
`therefore underset(x to0^(+))lim f(x)=underset(x to 0^(+))lim(|x|)/(x)`
`=underset(x to 0)limx/x" "...[ x to 0, x ne0]=1`
For `x le 0, |x|=-x`
`therefore underset(x to0)limf(x)=underset(xto0)lim(|x|)/(x)`
`=underset(xto0)lim(-1)" "...[x to0, xne0]=-1`
`thereforeunderset(xto0^(+))limf(x)neunderset(xto0^(-))limf(x)`
`thereforef` is discontinous ast `x=0.`
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