I'm not sure how far back you need to go. However, if you look at 4.3.16a and go from there, the authors claim that
This is most definitely true if it is known that is bounded. is called the stream function. If it represents a physical parameter, then most likely you can assume is bounded on physical grounds. The same goes for itself. But now, if and are both bounded, then it is the case that .
The authors also claim that goes to as goes to infinity. If that is the case, then if you take limits in the entire Equation 4.3.16a, you will get on the LHS, which implies you must get on the RHS. Hence, . Multiplying through by gives you Equation 4.3.16b.
So, in getting 4.3.16b from 4.3.16a, the problem reduces down to knowing that and are both bounded, and that .
With regard to the second derivative, you might be able to get that from the "note" in-between Equations 4.3.10 and 4.3.11: "... noting that ..." This tells you that , and hence . Is it the case that ?
These are just the beginnings of some ideas.