Let f:(0,∞)×R→R be 1-periodic in the second variable and in L∞((0,∞)×R). If it is necessary, we can also assume f to be continuous.
Suppose that f(t,x)→a∈R in L∞ on compact sets as t→∞. Do we have that f(t/ϵ,x/ϵ2)→a in L∞ on compact sets as ϵ→0?
Suppose that f(t/ϵ,x/ϵ2)→a∈R as ϵ→0 in L∞ on compact sets. Do we have that f(t,x)→a in L∞ on compact sets as t→∞?
If 1. and 2. are not true, is there a reasonable set of assumptions that make the statements true?