Let's push the limits!

Calculus Level 5

\[\large \lim_{x \to 0} \dfrac{(1+x)^{\frac{1}{x}}-e+ \frac{1}{2} ex}{x^2 (1+x)^{\frac{1}{x}}}\]

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

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

Clarification: \(e\approx 2.71828\) denotes Euler's number.


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