x->0
(1+2x)^(1/x)
=e^[ln(1+2x)/x]
=e^{[2x-(1/2)(2x)^2+o(x^2)]/x}
=e^[2-2x^2+o(x^2)]
√(1-2x)
=1-(1/2)(2x)-(1/8)(2x)^2 +o(x^2)
=1-x-(1/2)x^2 +o(x^2)
x-√(1-2x) =-1+(1/2)x^2 +o(x^2)
e^2.[x-√(1-2x)] =e^2.[-1+(1/2)x^2 +o(x^2)]
(1+2x)^(1/x)+ e^2.[x-√(1-2x)]
~e^2. { e^[-(2x^2+o(x^2)] -1+(1/2)x^2 }
~e^2. [ -2x^2 +(1/2)x^2 ]
=e^2. [ -(3/2)x^2 ]
lim(x->0) [(1+2x)^(1/x)+ e^2.[x-√(1-2x)] ]/x^2
=lim(x->0) e^2. [ -(3/2)x^2 ]/x^2
=-(3/2)e^2
(1+2x)^(1/x)
=e^[ln(1+2x)/x]
=e^{[2x-(1/2)(2x)^2+o(x^2)]/x}
=e^[2-2x^2+o(x^2)]
√(1-2x)
=1-(1/2)(2x)-(1/8)(2x)^2 +o(x^2)
=1-x-(1/2)x^2 +o(x^2)
x-√(1-2x) =-1+(1/2)x^2 +o(x^2)
e^2.[x-√(1-2x)] =e^2.[-1+(1/2)x^2 +o(x^2)]
(1+2x)^(1/x)+ e^2.[x-√(1-2x)]
~e^2. { e^[-(2x^2+o(x^2)] -1+(1/2)x^2 }
~e^2. [ -2x^2 +(1/2)x^2 ]
=e^2. [ -(3/2)x^2 ]
lim(x->0) [(1+2x)^(1/x)+ e^2.[x-√(1-2x)] ]/x^2
=lim(x->0) e^2. [ -(3/2)x^2 ]/x^2
=-(3/2)e^2