Let $X$ be a Banach space, let $\Omega\subset X$ be an open subset, and let $f:\Omega\rightarrow\mathbb{R}$ be a function such that the directional derivatives $$ \left.\frac{d}{dt}f(x+ty)\right|_{t=0} $$ exist for all $x\in\Omega$ and $y\in X$.
For fixed $x,y,z\in X$, what conditions must be satisfied for us to be able to be able to switch the derivative and limit in the following? $$ \lim_{\varepsilon\rightarrow 0^+}\left.\frac{d}{dt}f(x+ty+\varepsilon z)\right|_{t=0} = \left.\frac{d}{dt}\lim_{\varepsilon\rightarrow 0^+}f(x+ty+\varepsilon z)\right|_{t=0} $$