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Question

If f(x)=x1+exp(1/x),x00,x=0, then f(x) at x=0 is

A
Continuous
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B
Not continuous
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C
Differentiable
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D
Not differentiable
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Solution

The correct options are
A Continuous
C Not differentiable
Given f(x)=x1+exp(1/x),x00,x=0,
Continuity at x=0
LHL=limh00h1+e1/(0h)
=limh0h1+e1/(h)=01=0
=limh0h1+e1/(h)=01+=0
LHL=RHL=f(0)=0
Hence it is continuous
Differentiability at x=0
LHD=limh0f(0h)f(0)0h=limh00h1+e1/(0h)0h=1
RHD=limh0f(0+h)f(0)0+h=limh00+h1+e1/(0+h)0h=0
LHDRHD
Hence it is not differentiable at x=0

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