Let \(x_0\) be the real number such that \(e^{x_0}+x_0=0\). For a given real number \(\alpha\), define \(g\left(x\right)…

For \(\alpha =2,\lim _{x\to {x}_{0}}\left|\frac{g(x)+{e}^{{x}_{0}}}{x-{x}_{0}}\right|=0\)

For \(\alpha =2,\lim _{x\to {x}_{0}}\left|\frac{g(x)+{e}^{{x}_{0}}}{x-{x}_{0}}\right|=1\)

For \(\alpha =3,\lim _{x\to {x}_{0}}\left|\frac{g(x)+{e}^{{x}_{0}}}{x-{x}_{0}}\right|=0\)

For \(\alpha =3,\lim _{x\to {x}_{0}}\left|\frac{g(x)+{e}^{{x}_{0}}}{x-{x}_{0}}\right|=\frac{2}{3}\)