Question

Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b.

Does the limit of

$\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$

Exist as epsilon tends to 0? I think it is equal to the limit of

$\varepsilon \int_0^{1/\varepsilon} F(as) ds$

The latter limit does exist.

But I cannot prove it. ANY help would be appreciated.

Answer 1

Take $a=(1,0)$, $b=(0,1)$, $F(x,y)=sin(y)$. Then $\int_0^{1/\epsilon} F(as+b/\epsilon) ds= (1/\epsilon) \sin(1/\epsilon)$, so you are asking for the limit as $\epsilon$ goes to $0$ of $\sin(1/\epsilon)$. This limit does not exist.