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Question

limx0(1+x)1/xex

A
13e
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B
12e
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C
15e
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D
23e
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Solution

The correct option is B 12e
limx0(1+x)1/xex
Using L-hospital rule,

=limx0ddx(1+x)1/x

Let y=(1+x)1/x

logy=1xlog(1+x)

1ydydx=[x1+xlog(1+x)]x2

dydx=(1+x)1/x[x1+xlog(1+x)]x2

Therefore,

limx0(1+x)1/xex=limx0(1+x)1/e[x1+xlog(1+x)]x2

=e×limx0[x1+xlog(1+x)]x2

Again using L-hospital rule,

=e×limx0[(1+x)x(1+x)21(1+x)]2x

=e×limx0[1(1+x)21(1+x)]2x

Again using L-hospital rule

=e×limx0[2(1+x)3+1(1+x)2]2

=e×(2+12)=12e
Using L-hospital rule,

=limx0ddx(1+x)1/x

Let y=(1+x)1/x

logy=1xlog(1+x)

1ydydx=[x1+xlog(1+x)]x2

dydx=(1+x)1/x[x1+xlog(1+x)]x2

Therefore,

limx0(1+x)1/xex=limx0(1+x)1/e[x1+xlog(1+x)]x2

=e×limx0[x1+xlog(1+x)]x2

Again using L-hospital rule,

=e×limx0[(1+x)x(1+x)21(1+x)]2x

=e×limx0[1(1+x)21(1+x)]2x

Again using L-hospital rule

=e×limx0[2(1+x)3+1(1+x)2]2

=e×(2+12)=12e

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