  Given a $\sigma$-finite measure space $(X,\mu)$, a Young function $\Phi$, and
a one-parameter family of Young functions $\{\Psi_q\}$, we find necessary and
sufficient conditions for the associated Orlicz norms of any function $f\in
L^\Phi(X,\mu)$ to satisfy \[
  \lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}.
\] The constant $C$ is independent of $f$ and depends only on the family
$\{\Psi_q\}$. Several examples of one-parameter families of Young functions
satisfying our conditions are given, along with counterexamples when our
conditions fail.
