Nagel, Rudin and Shapiro (1982) investigated the tangential 
boundary behavior of the Poisson integrals of functions 
in the potential space 
$L^p_K(\mathbb R^n)=\{K*F:F\in L^p(\mathbb R^n) \}$ 
with a kernel $K:\mathbb{R}^n\setminus\{0\}\to[0,+\infty)$ 
which is positive, integrable, radial and decreasing.
In this paper, we extend the result %in \cite{NRS} 
to $L^{\Phi}_K(\mathbb R^n)=\{K*F: F\in L^{\Phi}(\mathbb R^n)\}$, 
where $L^{\Phi}(\mathbb R^n)$ is the Orlicz space.
Moreover we introduce $\Omega_R$-limit for a continuous increasing 
function $R:[0,+\infty)\to[0,+\infty)$ with $R(y)\to0$ as $y\to0$. 
The tangential approach region is defined by the function $R$. 
We give a relation between $R(y)$ for which all functions in 
$L^{\Phi}_K(\mathbb R^n)$ have the $\Omega_R$-limit
and the $L^{\widetilde\Phi}$-norm of $P_y*K$,
where $\widetilde\Phi$ is the complementary function of $\Phi$,
and calculate $R(y)$ precisely.