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Question

The function f : R/{0}R given by f(x)=1x2e2x1 can be made continuous at x=0 by defining f(0) as -

A
2
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B
1
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C
0
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D
1
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Solution

The correct option is D 1
f(x)=1x2e2x1=e2x12xx(e2x1)

Using expansion of ex

f(x)=(1+2x+(2x)22!+(2x)33!+...)12xx(e2x1)
f(x)=(2x)22!+(2x)33!x(e2x1)=2x((2x)2!+(2x)23!)x×(e2x1)
f(x)=2x(e2x1)×((2x)2!+(2x)23!)x
Now,
limx02x(e2x1)×(2x)2!+(2x)23!+...x=1×1=1
Hence, option D.

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