I am working through some exercises in Arnold's Mathematical Methods of Classical Mechanics book, specifically the second problem on page 20. For context, $T(E)$ is the period of motion along a closed phase curve corresponding to the energy level $E$. If $x_1$ and $x_2$ are where the phase curve intersects the $x$-axis (in the $x\dot{x}$-plane), then I've shown that the period of motion is $$T(E)=2\int_{x_1}^{x_2}\frac{1}{\sqrt{2(E-U(x))}}\,\text{d}x.$$ As for the problem, it says:
Let $E_0$ be the value of the potential function at a minimum point $\xi$. Find the period $T_0=\lim_{E\rightarrow E_0} T(E)$ of small oscillations in a neighborhood of the point $\xi$. The answer is $2\pi/\sqrt{U''(\xi)}$.
In order to do this I wanted to examine the period of the curve corresponding to energies $E$ close to $E_0=U(\xi)$. So I started with the equation $\frac{\dot{x}^2}{2} + U(x) = E$ and Taylor expanded $U$ about $\xi$. After rearranging the result, I got $$\dot{x}^2+U''(\xi)(x-\xi)^2=2(E-U(\xi)),$$ which is just an ellipse centered at $(\xi,0)$, whose semi-minor axis (along $x$-axis) has length $$\sqrt{\frac{2(E-U(\xi))}{U''(\xi)}}.$$ So then the period of the orbit on this curve is given by $$T(E)=2\int_{x_1(E)}^{x_2(E)}\frac{1}{\sqrt{2(E-U(x))}}\,\text{d}x,$$ where $x_1(E) = \xi-\sqrt{\frac{2(E-U(\xi))}{U''(\xi)}}$ and $x_2(E) = \xi+\sqrt{\frac{2(E-U(\xi))}{U''(\xi)}}$. Appealing to the integral form of the mean value theorem, this becomes $$T(E) = \frac{2(x_2(E)-x_1(E))}{\sqrt{2(E-U(\eta))}} = 4\sqrt{\frac{2(E-U(\xi))}{U''(\xi)}}\frac{1}{\sqrt{2(E-U(\eta))}}$$ where $\eta$ is some value between $x_2(E)$ and $x_1(E)$. However now if I try to take this limit as $E\rightarrow E_0$, then $x_1(E),x_2(E)\rightarrow \xi$ and $\eta\rightarrow \xi$ as well, so I get the result $$T_0 = \lim_{E\rightarrow E_0} T(E) = \frac{4}{\sqrt{U''(\xi)}}.$$ Which is off by a factor of $\pi/2$, and I cannot understand where the $\pi$ is even coming from. I included the tag for Perturbation theory because I'm not sure if there is some well known approximation that I am missing.