Suppose we have the ODE ($N \ge 2$):
$$
-\frac{1}{(\psi(r))^{N-1}}\left[(\psi(r))^{N-1} u^{\prime}(r)\right]^{\prime}=f(u(r))
\tag{0}
$$
Then we will get
$$
(\psi(r))^{N-1} u^{\prime}(r)=-\int_{r_0}^r(\psi(s))^{N-1} f(u(s)) \mathrm{d} s \tag{1}
$$
Then if $\psi(r) \rightarrow+\infty$, and the RHS of (1) is divergent, can we use L'Hôpital's Rule ?
If I use it then,
$$
\lim_{r \rightarrow +\infty} u^{\prime}(r)= \lim_{r \rightarrow +\infty} \frac{ - (\psi(r))^{N-1} f(u(r))} {(N-1)(\psi(r))^{N-2}\psi^{\prime}}
\tag{2}
$$
Did I misunderstand something ? Thank you very much ！
