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Question

If f(x)=x sgn(x1), then

A
f(x) is continuous at x=1
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B
f(x) is discontinuous at x=1
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C
limx1+x sgn(x1)=1
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D
limx1x sgn(x1)=1
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Solution

The correct option is D limx1x sgn(x1)=1

Given : f(x)=x sgn(x1)
we know,
sgn(x)=1x<00x=01x>0
So, only critical point is x=1

L.H.L.=limx1x sgn(x1) =limh0(1h) sgn(1h1)
=limh0(1h) sgn(h) =1×(1)
=1
R.H.L.=limx1+x sgn(x1) =limh0(1+h) sgn(1+h1)
=limh0(1+h) sgn(h)=1×1
=1
L.H.L.R.H.L.
f(x) is discontinuous at x=1.

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