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If f(x)={{:((x-|x|)/(x)","xne0),(2", ...

If `f(x)={{:((x-|x|)/(x)","xne0),(2", "x=0):},`show that `lim_(xto0) f(x)` does not exist.

Text Solution

Verified by Experts

L.H.L of `f(x)` at `x=0` is
`underset(xto0)limf(x)=underset(hto0)limf(0-h)=underset(hto0)lim(-h-|-h)/((-h))`
`=underset(hto0)lim(-h-h)/(-h)=underset(hto0)lim(-2h)/(-h)=underset(hto0)lim2=2`
R.H.L of `f(x)` at `x=0` is
`underset(hto0)limf(x)=underset(hto0)limf(0+h)=underset(hto0)lim(h-|h)/((h))`
`underset(hto0)lim(h-h)/(h)=underset(hto0)lim0/h=underset(hto0)lim0=0`
Clearly, `underset(xto0^(-))limf(x)neunderset(xto0^(+))limf(x)`
So, `underset(xto0^(-))limf(x)` does not exist.
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