Answer
$$\lim_{z\to0^+}\frac{2e^{1/z}}{e^{1/z}+1}=2$$
Work Step by Step
$$A=\lim_{z\to0^+}\frac{2e^{1/z}}{e^{1/z}+1}$$
Divide both numerator and denominator by $e^{1/z}$:
$$A=\lim_{z\to0^+}\frac{2}{1+\frac{1}{e^{1/z}}}$$
$$A=\frac{2}{1+\lim_{z\to0^+}\frac{1}{e^{1/z}}}$$
As $z\to0^+$, $e^{1/z}\to\infty$, so $1/(e^{1/z})$ will approach $0$.
$$A=\frac{2}{1+0}=2$$
