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.