Mathematics - Limits Question with Solution | TestHub

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