If α= lim_x → 0^+ ((e^√(tan x)-e^√ x)/(√(tan x)-√ x)) and β= lim_x → 0(1+ sin x)^1/2 cot x are the roots of the quadratic equation a x^2+b x-√ e =0,…

If $\displaystyle \alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\displaystyle \beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $\displaystyle a x^2+b x-\sqrt{e}=0$, then $\displaystyle 12 \log _e(a+b)$ is equal to $\displaystyle \_\_\_\_$.