Limit of $L_p$ norms again

Limit of $L_p$ norms again

A follow-up question to the recent problem.
Let $(E,\mathcal{M},\mu)$ be a measure space with $\mu(E) = \infty$. Let
function $f$ be such that for every $p\geq1$, $f \in L^\infty(E)\cap
L^p(E)$. Prove or find a counterexample
$$ \lim_{p\to\infty} \|f\|_p = \lim_{p\to\infty}
\frac{\|f\|_{p+1}^{p+1}}{\|f\|_p^p} = \|f\|_\infty. $$