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Question

Show that following function f(x), does not continuous at x=0
f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪e1x1+e1xx00x=0

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Solution

f(x)=⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪e1x1+e1xx0x=0

For continuity at x=0

f(0)=f(0)=f(0+)
f(0+)=limx0+e1/x1+e1/x

Expanding e1/x=1+1x+1x22!+.....

f(0+)limx0+1+1x+1x22!+......2+1x+1x22!+....

Applying LHospital Rule

f(0+)limx01x2+2x32!+.....1x2+2x32!+.....=1

f(0)f(0+)

Function is not continuous at x=0


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