First variation of a functional $J(y)$ 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
$$
J_{x,v}'(0)=D_vJ(x).
$$
