Solution
= Solution
\b[False.] Let
$$
k=\mathbf1_{\bigcup_{n\geq1}[n,n+2^{-n-2}]}.
$$
This is a bounded <measurable function>, and the intervals are disjoint, so
$$
\int_0^\infty|k(x)|\,dx=\sum_{n\geq1}2^{-n-2}<\infty.
$$
Nevertheless $k(n)=1$ for every positive <integer> $n$, while $k(n+1/2)=0$. Hence it has no limit at infinity. Narrow spikes separate <Lebesgue integrability> from pointwise decay.