I have this condition $$\lim\limits_{t\to0}\dfrac{f(x,t)}{t\phi(x,t)}=0\,\text{uniformly}\, x\in \mathbb{R}^N$$
can i deduce by using the definition of the limit that
$$\forall \varepsilon>0, \exists \delta>0, \forall t\in\mathbb{R}, |t|<\delta\Rightarrow |f(x,t)|\leq \varepsilon |t||\phi(x,t)|,\, \forall x\in\mathbb{R}^N$$