Using L-hospital rule,
=limx→0ddx(1+x)1/x
Let y=(1+x)1/x
logy=1xlog(1+x)
1ydydx=[x1+x−log(1+x)]x2
dydx=(1+x)1/x[x1+x−log(1+x)]x2
Therefore,
limx→0(1+x)1/x−ex=limx→0(1+x)1/e[x1+x−log(1+x)]x2
=e×limx→0[x1+x−log(1+x)]x2
Again using L-hospital rule,
=e×limx→0[(1+x)−x(1+x)2−1(1+x)]2x
=e×limx→0[1(1+x)2−1(1+x)]2x
Again using L-hospital rule
=e×limx→0[−2(1+x)3+1(1+x)2]2
=e×(−2+12)=−12e
Using L-hospital rule,
=limx→0ddx(1+x)1/x
Let y=(1+x)1/x
logy=1xlog(1+x)
1ydydx=[x1+x−log(1+x)]x2
dydx=(1+x)1/x[x1+x−log(1+x)]x2
Therefore,
limx→0(1+x)1/x−ex=limx→0(1+x)1/e[x1+x−log(1+x)]x2
=e×limx→0[x1+x−log(1+x)]x2
Again using L-hospital rule,
=e×limx→0[(1+x)−x(1+x)2−1(1+x)]2x
=e×limx→0[1(1+x)2−1(1+x)]2x
Again using L-hospital rule
=e×limx→0[−2(1+x)3+1(1+x)2]2
=e×(−2+12)=−12e