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If f(x)=sgn(x)" and "g(x)=x^(3),then pro...

If `f(x)=sgn(x)" and "g(x)=x^(3)`,then prove that `lim_(xto0) f(x).g(x)` exists though `lim_(xto0) f(x)` does not exist.

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`f(x)=sgn(x)={{:(1", "xgt0),(-1", "xlt0),(0", "x=0):}`
`:." "underset(xto0^(+))limf(x)=1" and "underset(xto0^(+))limf(x)=-1`
Now `f(x)xxg(x)={{:(x^(3)", "xgt0),(-x^(3)", "xlt0),(0", "x=0):}`
`:." "underset(xto0^(+))limf(x).g(x)=underset(xto0^(+))limx^(3)=0`
and`" "underset(xto0^(-))limf(x).g(x)=underset(xto0^(+))lim(-x^(3))=0`
Thus`" "underset(xto0)limf(x).g(x)` exists.
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