  We study Fourier convolution operators $W^0(a)$ with symbols equivalent to
zero at infinity on a separable Banach function space $X(\mathbb{R})$ such that
the Hardy-Littlewood maximal operator is bounded on $X(\mathbb{R})$ and on its
associate space $X'(\mathbb{R})$. We show that the limit operators of $W^0(a)$
are all equal to zero.
