Limits Continuity And Differentiability

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