Note: Expansion of (1+x)1/x=e(1−x2+11x224+o(x3))
∴limx→0(1+x)1/x−e+12exx2=limx→0e(1−x2+11x224+o(x3))−e+12exx2
=limx→0e−ex2+11ex224+o(x3)−e+12exx2
=limx→011ex224+o(x3)x2
=limx→011e24+o(x)
=11e24
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Where o(xn) is used to denote a power series where the least power of x is n.