Consider the difference equation $f_{x+1}-f_x=a(f_x)$ and the differential equation $g'_x=a(g_x)$. When and Why is it justified to say "$f_x - g_x = o(1) $ hence we can solve the difference equation with the differential equation" (not a quote). As an example see the answer I received here where $a:=ln$ .
An issue with approximations of a recurrence sequence
and the comment I gave to it.