$\begin{aligned} & \lim _{x \rightarrow 0} \frac{e^{\tan x}-e^x}{\tan x-x}=\lim _{x \rightarrow 0} \frac{e^x\left[e^{\tan x-x}-1\right]}{\tan x-x} \\ & =\lim _{x \rightarrow 0} e^x \cdot \lim _{x \rightarrow 0} \frac{e^{\tan x-x}-1}{\tan x-x}=e^0 \times 1=1 .\end{aligned}$