Some basic question: The first variation of a functional $J(y)$ is defined to be (see here, f.e.) $$ \delta J(y,h)=\lim_{\varepsilon\to 0}\frac{J(y+\varepsilon h)-J(y)}{\varepsilon}=\frac{d}{d\varepsilon}J(y+\varepsilon h)_{|\varepsilon=0}. $$
Hope this question is not too silly, but why do we have that $$ \lim_{\varepsilon\to 0}\frac{J(y+\varepsilon h)-J(y)}{\varepsilon}=\frac{d}{d\varepsilon}J(y+\varepsilon h)_{|\varepsilon=0}? $$
This is just an alternative definition. Consider the function $J_{x,v}\colon (-\varepsilon,\varepsilon)\to\mathbb{R}, t\mapsto J(x+tv)$. Then $J_{x,v}(0)=J(x)$ and $$ J_{x,v}'(0)=D_vJ(x). $$