Let's push the limits!

Calculus Level 5

limx0(1+x)1xe+12exx2(1+x)1x\large \lim_{x \to 0} \dfrac{(1+x)^{\frac{1}{x}}-e+ \frac{1}{2} ex}{x^2 (1+x)^{\frac{1}{x}}}

If the limit above exists and is of the form ab \dfrac ab, where aa and bb are coprime positive integers, find a+b.a+b.

If however, the limit doesn't exist or is irrational, enter 666 as your answer.

Clarification: e2.71828e\approx 2.71828 denotes Euler's number.

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