Regd. derivation of some equations in "Bertrand Spacetimes" by Pelick We are going through "Bertrand Spacetimes" by Dr Perlick, in which he first gave the idea of a new class of spacetimes named as Bertrand spacetimes after the well-known Bertrand's Theorem in Classical Mechanics.  
We could reproduce almost all of the equations in the paper, following the steps mentioned by Dr. Perlick, except for the two equations mentioned below:
\begin{align*}
{\beta}L_0={R_0}^2 \sqrt{U''(R_0)}\\
2{U_3}^2+2{R_0}^{-1}U_3+{R_0}^{-2}=U_4 - \frac{1}{2}{U_3}^2
\end{align*}
The paper says these equations were derived by taking the second and fourth derivatives respectively of the following equation in the limit $x{\rightarrow}0$:
\begin{align*}
(R_0-f(x))^{-1}-(R_0+x)^{-1}=\frac{2}{{\beta}L_0}\sqrt{2U(R_0+x)}
\end{align*}
Some results are also used in getting to these equations:
\begin{align*}
&f(x){\rightarrow}0\\
&f'(x){\rightarrow}1\\
&f''(x){\rightarrow}4U_3\\
&f^{(3)}(x){\rightarrow}24 {U_3}^2\\
&f^{(4)}(x){\rightarrow}48(8{U_3}^{3}-4U_3U_4+U_5)
\end{align*}
where
\begin{align*}
U_n:=\frac{U^{(n)}(R_0)}{n! U''(R_0)}
\end{align*}
Please guide so that I can get the equation and understand the paper.
 A: Here are some of the calculations I have done:
I am getting the second derivative without any limit as
\begin{align}
\frac{f''(x)}{(R_0-f(x))^2}+\frac{2 f'(x)^2}{(R_0-f(x))^3}-\frac{2}{(R_0+x)^3} = \frac{2 U(R_0+x) U''(R_0+x)-U'(R_0+x)^2}{\sqrt{2} \beta  L_0
   U(R_0+x)^{3/2}}
\end{align}
After putting the given limits, I get:
\begin{align} \frac{2 U^{(3)}(R_0)}{3 R_0^2 U''(R_0)}=\frac{2 U(R_0) U''({R_0})-U'({R_0})^2}{\sqrt{2} \beta  {L_0} U({R_0})^{3/2}} \end{align}
Similarly, for the fourth derivative, I get:
\begin{align}
\frac{f^{(4)}(x)}{({R_0}-f(x))^2}+\frac{6 f''(x)^2}{({R_0}-f(x))^3}+\frac{24
   f'(x)^4}{({R_0}-f(x))^5}+\frac{8 f^{(3)}(x) f'(x)}{({R_0}-f(x))^3}+\frac{36 f'(x)^2
   f''(x)}{({R_0}-f(x))^4}-\frac{24}{({R_0}+x)^5}=\frac{(-15 U'({R_0}+x)^4+4 U({R_0}+x)^2 \left(2 U({R_0}+x) U^{(4)}({R_0}+x)-3
   U''({R_0}+x)^2\right)-16 U({R_0}+x)^2 U^{(3)}({R_0}+x) U'({R_0}+x)+36 U({R_0}+x)
   U'({R_0}+x)^2 U''({R_0}+x))}{4 \sqrt{2} \beta  {L_0} U({R_0}+x)^{7/2}}
\end{align}
After putting the given limits, I get:
\begin{align}
\frac{6 \left({R_0}^2 U^{(5)}({R_0})+60 U^{(3)}({R_0})\right) U''({R_0})+20 {R_0}
   U^{(3)}({R_0}) \left((8 {R_0}+6) U^{(3)}({R_0})-{R_0} U^{(4)}({R_0})\right)}{15
   {R_0}^4 U''({R_0})^2}=\frac{(-15 U'({R_0})^4+4 U({R_0})^2 \left(2 U({R_0}) U^{(4)}({R_0})-3
   U''({R_0})^2\right)-16 U({R_0})^2 U^{(3)}({R_0}) U'({R_0})+36 U({R_0})
   U'({R_0})^2 U''({R_0}))}{4 \sqrt{2} \beta  {L_0} U({R_0})^{7/2}}
\end{align}
The Perlick paper also gives additional limits on U(R) and U'(R). $U(R){\rightarrow}0$ and $U'(R){\rightarrow}0$ in the limit $R{\rightarrow}R_0$. But with these conditions, both the equations take indeterminant form.
