Let $f: (0,\infty)\times \mathbb {R} \to \mathbb{R}$ be $1$-periodic in the second variable and in $L^\infty((0,\infty)\times \mathbb{R}).$ If it is necessary, we can also assume $f$ to be continuous.

1. Suppose that $f(t,x) \to a \in \mathbb{R}$ in $L^\infty$ on compact sets as $t \to \infty$. Do we have that $f(t/\epsilon, x/\epsilon^2) \to a$ in $L^\infty$ on compact sets as $\epsilon \to 0$?

2. Suppose that $f(t/\epsilon, x/\epsilon^2) \to a \in \mathbb{R}$ as $\epsilon \to 0$ in $L^\infty$ on compact sets. Do we have that $f(t,x) \to a$ in $L^\infty$ on compact sets as $t \to \infty$?

3. If 1. and 2. are not true, is there a reasonable set of assumptions that make the statements true?