Limit, Continuity and Differentiability: JEE Main 2024 Mathematics question

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 _________.