limit question on Lebesgue functions Let $f\in L^1(\mathbb{R})$.  Compute $\lim_{|h|\rightarrow\infty}\int_{-\infty}^\infty |f(x+h)+f(x)|dx$
If $f\in C_c(\mathbb{R})$ I got the limit to be $\int_{-\infty}^\infty |f(x)|dx$.  I am not sure if this is right.

Hint: if $f \ne 0 $ for $ x \in (a,b)$, and $h$ is large enough, $|f(x+h)+f(x)| \ne 0$ for $x \in (a,b) \cup (a-h,b-h)$. 
